Math 54 Mon-Fri 10-12 in 2 Evans

I'm grading the finals now. I will be done this weekend, and will submit grades by Monday. I have to give the final exams to the department. I believe you can look at them starting around the third week of classes in September - you just go to 970 Evans and talk to Marsha.

Like I said, if you want to come by my office and talk about math (or other things, I guess), feel free. I'm moving up to the 10th floor in a couple of weeks, but the change will be reflected on my website.

I had a really fun time teaching you guys this summer. I hope you enjoyed the class as much as I did. I think you all learned a lot. I know I certainly did. Good luck in your classes in the fall!

There will be a bonus problem on the final on Fourier series.

Here are some problems for you to work on. These are only from the later sections that weren't covered on the homework. Don't forget that many of the odd-numbered problems have solutions in the back of the book.

Section 9.6 # 1, 3, 5, 7, 15, 16

Section 9.8 # 1, 3, 5, 6

Section 10.3 # 1, 5, 9, 10, 12

The final exam is this Friday, August 14. Thursday's class will be a review session for the final exam. I will not prepare any material for the class. It is your responsibility to come prepared with questions to ask me. Office hours will be at the usual time on Thursday as well.

As I said in class, the exam will be cumulative, but will be about two thirds differential equations and one third linear algebra. You can expect the format and length to be similar to the previous tests. You will be responsible for everything we do in class up through Wednesday. Also, remember that definitions and statements of named theorems are fair game.

The following are the specific skills/procedures you may be tested on. Linear algebra stuff:

• Find a basis for the null space and column space of a given matrix. Determine the rank of the matrix and the dimension of its null space. Determine if the associated transformation is one-to-one or onto.
• Determine if a given matrix is invertible, and find its inverse if so.
• Given a square matrix A, find its characteristic polynomial, eigenvalues, and a basis for each eigenspace. Diagonalize the matrix if possible (i.e. find the P and the D) or else explain why it is not diagonalizable.
• Orthogonally diagonalize a symmetric matrix.
• Using the Gram-Schmidt process to find an orthogonal basis of a subspace of a vector space.
• The following basic linear algebra stuff is fair game too: row-reduction, solving a linear system (homogeneous or non-homogeneous), determining dependence/independence of a set of vectors, determining if something is a subspace of a vector space, computing determinants.

Differential equations stuff:

• Solve a second-order linear constant-coefficient homogeneous DE. You should be able to solve a third-order one if the auxiliary equation is easy to factor.
• Find a particular solution to a second- or third-order linear constant coefficient nonhomogeneous DE using the method of undetermined coefficients.
• Solve an initial-value problem for either of the previous two situations.
• Find the interval on which a solution must exist for a DE with nonconstant coefficients (i.e. remember the Existence and Uniqueness Theorem). Make sure that you put the equation in standard form first.
• Use the Wronskian to determine linear dependence/independence of a set of solutions to a homogeneous linear DE, find a fundamental solution set.
• Convert a higher-order linear DE into a first-order linear system of DEs in normal form.
• Find solutions to a first-order constant-coefficient homogeneous linear system of DEs using the eigenvalue-eigenvector formula, when the coefficient matrix is diagonalizable.
• Find a fundamental matrix for a diagonalizable system using matrix exponentiation.
• Use the Wronskian to determine if a collection of solutions is linearly dependent/independent.
• Graph the trajectories of a 2x2 differential system. Identify sources, sinks, saddles, and spirals. Determine if a spiral is spiraling clockwise or counterclockwise, and if it is spiraling in or out.

This week we learned some more tools for solving differential equations, and also for understanding when solutions to homogeneous DEs are linearly independent. We finished by talking about first-order differential systems and their connection to higher-order DEs.

Day 1: Given two solutions to the DE ay''+by'+cy=0, we defined something called their Wronskian, which is the determinant of a matrix formed by the two solutions and their first derivatives. We saw that the Wronskian is either always zero or never zero; furthermore, it is zero if and only if the two solutions were linearly dependent (and hence it is nonzero if and only if the two solutions were linearly independent). This only needs to be checked at one point, provided that the two functions are solutions to the same homogeneous linear DE.

We also talked about the method of undetermined coefficients, which allowed us to find solutions to nonhomogeneous second-order linear constant-coefficient DEs. Basically, you look at the function on the right-hand side of the DE, and for your particular solution you make a guess which is of the same form as the right-hand side, but with variable coefficients. Then you compute the derivatives of your guess, plug them into the left-hand side of the equation, and obtain some linear equations involving the variable coefficients you introduced, then solve them to find the particular solution.

Day 2: We saw the method of undetermined coefficients sometimes doesn't work on the first try. But we also saw how to fix it: if the right-hand side of the DE is of the form p(t)e^rt, and r is a root of the auxiliary equation, then you multiply your guess by t. If r is a double root of the auxiliary equation, you have to multiply by t^2 instead of t. There was a similar rule for when the right-hand side is of the form cos(bt)p(t)e^at (or sin(bt)): if a+ib is a root of the auxiliary equation, multiply by t.

Day 3: We talked about superposition and the method of undetermined coefficients. This means that if your DE has multiple terms on the right-hand side, you can make a guess for each of them, solve the resulting linear systems separately, then combine your guesses. All of this stuff works for higher-order equations as well. We saw that, even if we can't factor the auxiliary equation for a higher-order DE, we can still determine if a given number r is a root of it; this is done just by plugging in r and seeing if you get 0.

Day 4: We moved on to Chapter 6, which was mainly theoretical, on higher-order linear differential equations, not necessarily homogeneous or with constant coefficients. We defined an nth order DE to be in standard form if it looked like y^(n)+p_0(x)y^(n-1)+...+p_(n-1)(x)y=g(x), where p_0,...,p_(n-1) and g are all functions of x. The big theorem is that, on an interval (a,b) containing x_0, there is always a unique solution to the differential equation satisfying any given set of initial conditions. These initial conditions are specified values for the value of y and its first (n-1) derivatives at the point x_0. This theorem requires that the coefficient functions are continuous on the interval (a,b).

This proves that if the DE is homogeneous, then the solution space is n-dimensional. We also had a theorem (similar to what we saw in linear algebra) that said that if the equation is not homogeneous, and if we have a particular solution y_p to the inhomogeneous equation, then every solution to the inhomogeneous equation is of the form y_p+c_1y_1+...+c_ny_n, where y_1,...,y_n are n linearly independent solutions to the associated homogeneous equation and c_1,...,c_n are constants. Again, we can use this form to find the solution to initial-value problems.

Day 5: We moved on to Chapter 9, which talks about linear systems of first-order differential equations (first-order differential systems) and their connection to higher-order linear differential equations. We saw that by introducing new functions, we could convert a higher-order linear differential equation in standard form into a first-order differential system in what is called normal form: this means the system is represented in the form x'=Ax+f, where x, x', and f are thought of as vectors of functions, and A is a matrix of functions. We had a theorem guaranteeing existence and uniqueness of solutions to these systems also, similar to previous existence and uniqueness theorems.

Finally, we talked about how to solve these first-order systems in the case when the matrix A was a constant matrix, and when the equation was homogeneous (i.e. in the form x'=Ax). We showed that we can find solutions of the form x(t)=(e^rt)u, where u is an eigenvector for A with eigenvalue r. That was cool. I talked a little bit about graphing the trajectories of solutions to initial-value problems, but didn't get into much detail. We will talk more about this next week.

This week we finished the linear algebra book and started differential equations. And of course there was the second midterm.

Day 1: We talked about inner products in general vector spaces. An inner product is a function that takes two vectors in V and gives you a real number. It has to satisfy the same properties as the dot product in R^n. Once you have one of these, you have all the same notions and theorems as you have in R^n. You can define orthogonality, orthogonal projection, Gram-Schmidt, the Pythagorean Theorem, etc. The main examples we saw were the evaluation inner product on P_n (remember, you need to pick n+1 points for P_n), and the integration inner product on C[a,b].

Day 2: We talked about diagonalization of symmetric matrices. A matrix A is symmetric if A=A^T. We started with an example; we diagonalized a symmetric matrix, and it turned out that the eigenvectors for different eigenvalues were orthogonal to each other. Then we proved a general theorem saying that this is always the case for a symmetric matrix. The next big theorem (which we didn't prove) said that a matrix is symmetric if and only if it is orthogonally diagonalizable. This means that the matrix P in the equation A = PDP^(-1) is in O(n), so its columns form an orthonormal basis for R^n consisting of eigenvectors of A. When we orthogonally diagonalize a matrix, we may have to do Gram-Schmidt to get an orthonormal basis in each eigenspace separately, but the first theorem guarantees that different eigenspaces will be orthogonal to each other already. Finally we had the Spectral Theorem, which listed a bunch of facts about symmetric matrices and their diagonalizability.

Day 3: We started talking about differential equations. We introduced the general second-order linear constant-coefficient homogeneous differential equation ay''+by'+cy=0. That's a lot of words: what do they all mean? Second-order means that y'' is the highest derivative involved; linear means that each derivative term appears by itself, not multiplied by another derivative term; constant-coefficient means that a,b,c are real numbers and not functions; and homogeneous means that the right-hand side of the equation is 0.

We saw how to solve these, at least in the cases when the discriminant b^2-4ac was greater than or equal to 0. You form the auxiliary equation ar^2+br+c=0, solve for r by factoring or using the quadratic formula, and then there is a formula for the solution to the DE which involves the exponentials of the roots of the auxiliary equation. The situation is different depending on whether the auxiliary equation has two distinct real roots, a repeated real root, or two complex conjugate roots. We dealt with the first two cases today.

Day 4: We dealt with the last case for the differential equation we studied the previous day. This is when the auxiliary equation has two complex conjugate roots. We talked about Euler's formula, which relates the exponential of an imaginary number to the sine and cosine functions. We saw that we could take (complex) linear combinations of the complex solutions to the DE (which were given by exponentials of complex numbers) and create real-valued solutions to the DE. That was neat.

Day 5: Second midterm.

Glen asked in class today what I would write about if I had to write a paragraph about something that I like, dislike, or am indifferent to. I said Canada, but I've reconsidered. Although I do love Canada. But I decided to write about something else. This is my website, I can do what I want. So here you go.

The thing I like more than anything else is traveling to other countries and seeing what things are like there. When I hear about a place in the news or in conversation, I have some picture in my mind of what it's like, how things look, how people dress, etc, but that picture is generally wrong. So I like to go visit different places and find out for real. The thing I like to do most when I go to a new country or city is just to walk the streets for a couple of days, to see different neighborhoods, the architecture, the street life. I think I enjoy traveling so much because it reminds me that, well, things are different in other places. It sounds dumb to write it like that, but I think it's refreshing for the soul (if there is such a thing) to be in a new environment sometimes. The most interesting trip (to me - it's ok if you're not interested) I've ever gone on was a recent trip to Iran in April and May of this year. The people there are friendlier and more welcoming to strangers than any I've met anywhere else in the world. I spent about three weeks there and I hope I get a chance to go back sometime.

Well, I'm sure you're aware that the second midterm is this Friday 7/31. The format will be quite similar to the first midterm. All definitions and named theorems are fair game, as are worksheet problems and homework problems. You don't need to know how to prove the theorems, but there may be some theoretical problems on the midterm, and a degree of comfort with the theoretical content of the course will definitely help you.

I will have extended office hours on Thursday, from 1:00 to 3:30. Definitely come by if you're having trouble with any of the material we've covered.

The second midterm will cover the following sections: 4.4-4.7, 5.1-5.4, 6.1-6.5, 6.7, and 7.1. There will be no material on differential equations on the second midterm. The following is a list of specific skills/procedures that you may be tested on:

• Given a basis for a vector space, find the coordinates of a vector relative to that basis; or, given a coordinate vector, find the vector with those coordinates.
• Use the coordinate mapping (relative to a basis) to determine whether a set of vectors is linearly independent or not.
• Compute a change of basis matrix in a general vector space, given two bases.
• Given a linear transformation T : V --> W, and bases for V and W, find the matrix of the linear transformation relative to those bases.
• Given a matrix A, determine if a given vector is an eigenvector, or if a given scalar is an eigenvalue.
• Compute the characteristic polynomial of a given matrix, and factor it to find the eigenvalues.
• Find a basis for the eigenspace corresponding to an eigenvalue of a given matrix.
• Diagonalize a given matrix, or explain why the matrix cannot be diagonalized.
• Compute dot products, norms, and distances in R^n. Determine if a set is orthogonal or orthonormal.
• Project a vector onto another vector (ie onto a line) in R^n. Write the original vector as the sum of a vector on the line and a vector orthogonal to the line.
• Project a vector onto an arbitrary subspace W of R^n. Write the vector as the sum of a vector in the subspace and a vector orthogonal to the subspace. Find the closest point in W to a given vector.
• Find the matrix corresponding to the linear transformation given by projecting a vector onto the subspace W.
• Given a basis for a subspace W of R^n, use the Gram-Schmidt algorithm to produce an orthogonal basis for that subspace.
• Do the previous five things in a general inner product space. Except for finding the matrix for the projection.
• Compute a QR factorization of a matrix with independent columns.
• Compute the set of least-squares solutions to a given linear system by writing the system of normal equations and solving it. Compute the least-squares error for a given system.
• Orthogonally diagonalize a symmetric matrix.

This week we covered sections 1 through 5 of Chapter 6. Chapter 6 started with the notion of dot product (which we talked about earlier, but not systematically), and we saw that the dot product leads to the notion of vectors being orthogonal (ie perpendicular) in R^n. A lot of useful concepts flow from this idea.

Day 1: Section 6.1 was on dot product, length, and orthogonality. We spent most of the day defining things and computing examples. The first thing was some properties of the dot product: the most important ones are (a) linearity in each variable, so (cu + dv)*w = c(u*w) + d (v*w) (I'm using the * to indicate dot product since I don't know how to make a big dot in html), and (b) positive definiteness, so u*u >= 0, and u*u = 0 if and only if u = 0.

We said that two vectors are orthogonal if their dot product is zero, and we defined the orthogonal complement of a set S to be the set of all vectors which are orthogonal to every vector in S. We showed that the orthogonal complement of any set of vectors is a subspace (remember, you can show this by just checking the three conditions), and we had a theorem (Theorem 3) that says the orthogonal complement of the row space of A is the null space of A, and the orthogonal complement of the column space of A is the null space of the transpose of A. Check it out for yourself - it all follows from the definitions of matrix multiplication.

We defined the length of a vector u to be the square root of u*u, and we also called that number the norm of u. We had a theorem that showed that orthogonal vectors obey the Pythagorean Theorem, which shows us that linear algebra is connected with things that people knew over 2500 years ago. Holy crap.

Day 2: Section 6.2 was about orthogonal (and orthonormal) sets of vectors. A set is orthogonal if each pair of distinct vectors in the set is orthogonal. The first thing we observed was that if we have a set of nonzero vectors which is orthogonal, then it must be a LI set. Next we showed that if you have an orthogonal basis for a subspace W, and you want to express a vector y in W in terms of those basis vectors, you can find the coefficients easily - the coefficient of u_i s (y*u_i)/(u_i*u_i). This works because dotting with u_i kills all the other basis vectors. See Theorem 5 on page 385.

Then we talked about the idea of orthogonal projection of a point onto a line (or onto another vector). Fix a nonzero vector u, and let L be the line spanned by u. Let y be any vector. Then we found a formula for writing y as a sum of a vector in the direction of u (ie a vector in L) and a vector perpendicular to u (ie perpendicular to L).

We defined an orthonormal set to be an orthogonal set consisting of unit vectors. This is like an orthogonal set, but the vectors must all have length one; so an orthogonal set can contain the zero vector, but an orthonormal set cannot. We showed that if U is an m by n matrix, then the columns of U are an orthonormal set if and only if (U^T)U = I. If this is the case then we must have n no greater than m (by independence of the columns). Then we showed that if we think of U as defining a transformation from R^n to R^m, it doesn't change dot products. Thus it doesn't change lengths or angles.

Day 3: Section 6.3 was about orthogonal projections. We did this before for one-dimensional subspaces, and we generalized it to subspaces W which are of arbitrary dimension. We need an orthogonal basis for W to actually compute projections. We showed that the projection of y onto W is given by projecting y onto each of the vectors in the orthogonal basis and then summing up the results. We showed that we can uniquely write y as the sum of a vector in W and a vector in the orthogonal complement of W. This told us that it didn't matter which orthogonal basis we chose for W - we would still get the same result for the piece of y that lives in W (although the projections onto the individual basis vectors would be different).

We showed that the projection of y onto W gives the closest point in W to y, and this tells us that if we took y to be in W to start with, then we'd get y itself back when we project. Finally, we had a theorem which said that if we take U to be a matrix whose columns form an orthonormal basis for W, then U(U^T) is the standard matrix of the linear map from R^n to R^n which takes a vector y and projects y onto W. ie U(U^T)y = proj_W(y). If we denote P = U(U^T), we saw that P^2 = P, and hence P has 0 and 1 as its only eigenvalues (as we saw on the quiz).

Day 4: Section 6.4 was about the Gram-Schmidt process, which is a procedure which takes a basis for a subspace W of R^n and produces an orthonormal basis from it. I won't write out the formula here, but it basically involved successively subtracting off projections of the original basis vectors onto the orthogonal basis vectors that you have already created. If that doesn't make sense, check your notes or read the book. In fact, I recommend that you check your notes and read the book anyway. We saw that Gram-Schmidt gave us a way to take a matrix A with LI columns and write it in the form A = QR, where Q has orthonormal columns and R is invertible, upper triangular, and has positive diagonal entries.

Day 5: Section 6.5 was about least-squares problems. In the real world, sometimes you have to try to solve a linear system for which there is no exact solution, ie the system is inconsistent. Then, what you do is you try to find the closest thing possible to a solution. We defined a least-squares solution to Ax=b to be a vector z (I used x with a hat over it in class) such that ||Az - b|| is minimized.

We made the observation that, whatever Az is, it lives in the column space of A, so we want to project b onto the column space of A and solve Az=(projection of b onto ColA). But that's a lot of work since we'd then have to compute an orthogonal basis for ColA.

Instead, we came up with a clever alternative solution. If the original system was Ax=b, we defined a new system (called the system of normal equations), which is (A^T)Ax = (A^T)b. This system always has solutions (which is not obvious), and the solution set of the system of normal equations is precisely the set of least-squares solutions to Ax=b. We ended with a couple of theorems about when the least-squares solution is unique (when the columns of A are LI), how to find the solution using [(A^T)A]^(-1), and how to find it using a QR factorization of A.

This week we did a lot of theoretical stuff. We finished Chapter 4, which was on abstract vector spaces and linear transformations between them, and we also did sections 1 through 4 on Chapter 5, which was on eigenvalues and eigenvectors.

I know this stuff is abstract and confusing - don't feel bad if you don't understand everything right away. It takes time and practice to become comfortable with all the new concepts and notation. This is true for me also. I've been studying math for 10 years now; when I'm taking a class, it's not like I understand everything instantly. The difference is that I'm more comfortable with the feeling of confusion that comes when somebody introduces a new idea. When you don't understand something, don't panic! Go home, read the textbook, try the problems, and see if you can relate the new stuff back to the stuff we studied before. If you still have trouble, come to office hours. OK, now on to the summary:

Day 1: Started by discussing the midterm problems. Then we did a bunch of stuff in Chapter 4 - sections 4.4 through 4.7, more or less. 4.5 was on the idea of dimension of an abstract vector space (which we saw previously in R^n), and we saw that things were defined exactly like they were before - the dimension of V is the size of any basis for V. We had a theorem that said that any two bases must have the same size - this is true because any time you have p vectors in an n-dimensional space, with p > n, they must be dependent (we saw this in R^n previously as well). 4.6 was on the idea of rank of a linear transformation, which is the dimension of the image of the transformation. We had a theorem which asserted that if T : V -> W is linear, the sum of the rank of T and the dimension of the kernel of T must equal dim(V). Again, this was familiar from our experience with matrices in R^n.

Sections 4.4 and 4.7 were on coordinate systems and change of coordinates. This stuff is a little trickier. A basis for a vector space V gives you a coordinate map, which is an isomorphism of V with R^n, where n is the dimension of V. This allows us to translate problems in V to problems in R^n, where we can use techniques we've already learned (mainly row-reducing matrices) to solve the problems, and then translate the results back to V using the inverse of the coordinate map. Change of coordinates shows us how to switch back and forth between different coordinate systems.

Day 2: We started Chapter 5 by introducing the idea of eigenvectors for a square matrix. If A is n by n, this defines a linear transformation from R^n to itself. The nonzero vectors v which are mapped to scalar multiples of themselves, ie Av = cv for some scalar c, are called eigenvectors of A. The number c is called the eigenvalue. Remember, 0 can be an eigenvalue, but the 0 vector cannot be an eigenvector.

We spent some time figuring out how to find eigenvalues and eigenvectors. We saw that c is an eigenvalue if and only if (A - cI) is not invertible (so it has some nonzero kernel, which is precisely the set of eigenvectors of A with eigenvalue c). Thus, c is an eigenvalue if and only if det(A-cI)=0. To check that v is an eigenvector for A just requires us to compute Av and see if it is a scalar multiple of v. We had a good theorem at the very end that said that eigenvectors for distinct eigenvalues must be linearly independent.

Day 3: We did section 5.2 today. The observation that c is an eigenvalue if and only if det(A-cI)=0 also gave us a procedure to find eigenvalues: thinking of c as a variable, we define the characteristic polynomial det(A - cI), find the values of c for which the polynomial equals 0 (by factoring), and these are the eigenvalues. Then to find the eigenvectors associated to the eigenvalue c, row-reduce the matrix (A - cI) and find a basis for its kernel, ie a basis for the solution set to the homogeneous system (A - cI)x=0. We also made another addition to the Invertible Matrix Theorem - A is invertible if and only if 0 is not an eigenvalue of A. Finally, we saw that if A has n eigenvalues, then det(A) is the product of those eigenvalues.

Day 4: Today we did section 5.3, on diagonalizing a matrix. We introduced the idea of similarity: A is similar to B if A = PBP^(-1) for some invertible matrix P. We defined A to be diagonalizable if A is similar to a diagonal matrix D, ie A = PDP^(-1). We had an important theorem that told us that an n by n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In that case, the eigenvectors of A are the columns of P (which must be LI since P is invertible) and the eigenvalue corresponding to the jth column of P is the (j,j) entry of the diagonal matrix D.

We saw how to diagonalize a matrix A in three steps: (i) Find the eigenvalues of A using the characteristic polynomial; (ii) Find the corresponding eigenvectors by row-reducing (A - cI) for each eigenvalue c, thus finding a basis for the eigenspace corresponding to c, and finally (iii) Write down the matrices P and D. If there are n eigenvectors, A is diagonalizable; if there are fewer than n eigenvectors, A is not diagonalizable. Finally, we saw that if A has n distinct eigenvalues, A must be diagonalizable (since each eigenvalue has at least one eigenvector associated with it, so there must be n of them).

Day 5: We did section 5.4 on Eigenvectors and Linear Transformations. This stuff is probably the hardest that we've done so far in the course. We put together some ideas we had seen before - coordinate systems in a general vector space, and linear transformations between general vector spaces. We saw that if T : V -> W is linear, and if we choose bases for V and W, then we get a matrix that represents T.

How does this work? Well, choosing a basis for V gives us an isomorphism of V with R^n, and similarly choosing a basis for W gives us an isomorphism of W with R^m. Then we can think of T as giving us a linear transformation from R^n to R^m, which we know must be given by a matrix. This matrix depends on both of the bases as well as the transformation. Doing this allows us to determine things about T (like the dimension of its image or its kernel) using techniques that we've already developed. The matrix was defined as follows: the jth column is the image of the jth basis vector of V, expressed in coordinates using the given basis for W. Read in your notes or your book if that doesn't make sense verbally.

We had a long, difficult deduction that (I claim) showed us that similar matrices represent the same transformation, but in different bases. To be more precise, suppose A = PCP^(-1). We have a transformation from R^n to R^n given by multiplication by A. Using the idea above, we can express this transformation in terms of a different basis, namely the basis given by the columns of P. The answer was that, with respect to that basis, the linear transformation is represented by C.

The way to think about this is as follows: A = PCP^(-1) expresses the equality of two linear transformations. We interpret P as the change of coordinates matrix from (coordinates given by columns of P) to standard coordinates, so then P^(-1) changes from standard coordinates to (coordinates given by columns of P). Then the transformation PCP^(-1) means: change from standard coordinates to (coordinates given by columns of P), do the transformation C, then change back to standard coordinates. The fact that PCP^(-1) = A means that these two transformations are equal. Like I said in class, roughly speaking this means that A and C represent the same transformation if you tilt your head to the side a little bit.

Homework 8 is now posted. It is due next Tuesday, July 21. Do it. It will be good for you.

We're halfway through the course already! It's gone by so quickly. At least, it has for me.

In case you missed it, Homework 6 is now posted on the calendar. It is due on Tuesday, July 14. We haven't quite covered all the material on it yet, but most of it is quite similar to what we've done before, so I think you'll be ok. Please come to office hours Monday if you need help. That's what I'm there for.

Also, it has come to my attention that Problem 20 from Section 2.3, on Homework 4, was graded somewhat harshly. If you feel that you lost points on this problem unfairly, please bring your homework to me on Monday and I will regrade that problem.

This week we finished Chapter 2 with the sections on subspaces, and dimension and rank. Then we moved on to Chapter 3 of Lay, on determinants.

Day 1: We talked about coordinate systems. The idea is this: when you choose a basis for a subspace H of a vector space V, every vector x in H can be expressed in a unique way as a linear combination of the basis vectors. The set of weights in that linear combination gives you a vector in R^p (where p is the number of elements of the basis), and this vector is called the coordinate vector of x. This formalizes the idea that a subspace with dimension p "looks like" R^p. We said that the coordinate mapping from H to R^p is one-to-one and onto, ie it is an isomorphism.

We had a very important theorem: the Rank-Nullity Theorem, which states that for a matrix A with m rows and n columns, the dimension of its column space and the dimension of its null space add up to n. Think about it this way: A defines a linear transformation from R^n to R^m, and Rank-Nullity says that the null space and the column space complement each other to account for all the dimensions of R^n, the domain of the transformation. A good example of this is the projection map from R^3 to R^2 sending a vector (x,y,z) to (x,y).

Day 2: We talked about determinants a lot. We defined what is a determinant for an n by n matrix, and we saw (in the 2x2 and 3x3 cases) how it comes from row-reducing a general matrix. The determinant is defined as an alternating sum of products of entries of the matrix with determinants of smaller sub-matrices, and then said that it could be computed by expanding down any column or across any row, as long as you keep track of signs correctly. Somebody made the smart observation that one should always choose the row or column with the most zeros in it.

Then we explored the effect of row operations on the determinant of a matrix. We concluded that replacing a row with the sum of itself plus a multiple of another row doesn't change the determinant, swapping two rows changes the sign of the determinant, and scaling a row by k changes the determinant by a factor of k. Thus, elementary row operations do not change whether or not the determinant is zero, although they may change its sign or its absolute value. Since A is invertible if and only if A is equivalent to the identity matrix, we concluded with the theorem we wanted, namely that A is invertible if and only if the determinant of A isn't zero.

Day 3: We had a few more properties of determinants. The most important one is that the determinant is multiplicative, ie the determinant of a product is the product of the determinants. We talked about how to use determinants to calculate area and volume. After doing some final problems on determinants, we moved on to Chapter 4: Vector Spaces!

We had an abstract definition of vector spaces, with 10 axioms that they had to satisfy. I said that these axioms are basically there so we know that everything behaves the way we have seen that things behave in R^n. We had some examples, such as spaces of sequences, spaces of polynomials, and spaces of continuous functions. We talked about subspaces of vector spaces. For example, P_3 (the space of polynomials of degree at most 3) is a subspace of P_5, but R^3 is not a subspace of R^5. Although this might sound confusing, it's pretty simple. The reason that R^3 fails to be a subspace of R^5 is that R^3 is not a subset of R^5; a vector with three coordinates is not a vector with five coordinates.

Day 4: We went quickly through a lot of stuff in Chapter 4. We had definitions of linear independence and bases in general vector spaces, which were identical to the definitions we had seen in R^n previously. Just relax! This stuff isn't scary. We defined linear transformations between vector spaces, and the kernel (like the null space) and range (like the column space) of a linear transformation, except that the transformation might not be defined by a matrix.

In our talk about bases, we said that there are two ways of thinking about a basis: on the one hand, it is the smallest possible (or most efficient) spanning set; on the other hand, it is the largest possible independent set. Theorem 5 in Section 4.3 is important. It says that if you have a set S of vectors in a vector space V, and H is the span of the vectors in S (so H is a subspace) then if one vector is a linear combination of the others, you can throw it out of the set and the remaining vectors will still span H. Furthermore, it says that some subset of S spans H; so anytime you have a spanning set, you can keep tossing out redundant vectors until you get a basis.

Day 5: midterm. If you have any comments about the midterm (too long, too short, too difficult, too easy, just right, etc), feel free to email me. I'd like your feedback. I will, of course, keep any comments you make confidential.

I'm sure you're aware that the first midterm is coming up this Friday. I have posted Homework 5 on the Google calendar. You don't have to hand it in, but the problems would be good to review for the midterm.

The midterm will cover everything up to what we do on Wednesday. I recommend going over the worksheet problems and the homework problems to study. I know there's a lot of material. Just remember, if you're stuck, don't panic. Look at the problem and ask yourself if you know what the words mean. What are the definitions? Also, remember that almost everything we've talked about boils down to something about existence or uniqueness of solutions to linear systems. Try to relate things back to that context.

Speaking of definitions, here's what I do if I want to remember a complicated definition: I write it out, by hand, like six times. Or however many times it takes me to remember it. This also works for theorems. I am not joking. I do this all the time.

I am planning to hold extra office hours on Thursday, from 1:00 to 3:30. You should definitely come by if you're having trouble with the material.

This week we finished up Chapter One of the textbook and moved into Chapter Two. Again, we spent a lot of time reformulating the concepts we've been learning about - namely solving linear systems and how to solve them - into more abstract questions about properties of matrices or sets of vectors in R^n.

Day 1: We started with the concept of linear transformations. We defined the domain, codomain, and range of a linear map. The range may also be called the image of the map. We saw that an m by n matrix gives a linear transformation from R^n to R^m, and that solving Ax = b is equivalent to finding the preimage of b under that transformation. We saw a lot of geometric examples of linear maps: scaling, rotation, shearing, and projection.

Day 2: We asked whether or not every linear transformation from R^n to R^m arises from an m by n matrix, and Theorem 10 showed that this is true. The really important fact here is that the columns of the matrix correspond to the images of the standard basis vectors for R^n (in order!). We defined what it means for a linear transformation to be onto (an existence question) and one-to-one (a uniqueness question). We showed that a linear transformation is one-to-one if and only if its null-space consists only of the zero vector. Moving on to Chapter 2, we talked about matrices and things you can do with them, namely adding, multiplying by scalars, and then matrix multiplication. We saw that matrix multiplication corresponds to composition of linear maps. Be cautious! Not everything that holds for multiplication of real numbers holds true for matrices. For example, AB = BA is not true for all matrices A and B (so matrix multiplication is not commutative) and AB = AC does not imply that B = C (so cancellation does not hold).

Day 3: We talked about transpose and about taking powers of square matrices before moving on to section 2.2, about invertible matrices. We said that A is invertible if there is a matrix C such that AC = I_n and CA = I_n (and we noted that A must be a square matrix). We showed that C is unique, and we called it the inverse of A. We said that A is called nonsingular or invertible if such a C exists, and singular otherwise. We showed that a 2 by 2 matrix is invertible if and only if its determinant is nonzero, and in that case we showed a simple formula to find the inverse. We then had the very important Theorem 5 which stated that if A is invertible, then the equation Ax=b is consistent for all b, and also that the solution is unique for all b. This means that A has a pivot position in each row and each column, so A is row-equivalent to the identity matrix. Finally we showed that if A is invertible, then so are A^(-1) and A^T, and if B is also invertible, so is AB, and the inverse of the product is the product of the inverses in reverse order.

Day 4: We defined elementary matrices as matrices that are obtained by performing a single elementary row operation on the identity matrix, and we showed that performing a row operation on a matrix A is equivalent to multiplying A on the left by E, where E is the elementary matrix corresponding to that row operation. In symbols, A ~ EA. Then Theorem 7 showed that A is invertible if and only if A ~ I_n, and in this case we can find A^(-1) by performing on I_n the same sequence of elementary row operations that we used to reduce A to I_n. Moving on to section 2.3, we saw a theorem (Theorem 8) giving twelve equivalent conditions for a matrix to be invertible. I didn't write them all down, but the really important fact here is that a square matrix A has a pivot position in every row if and only if it has a pivot position in every column. In other words, Ax = b is consistent for all b if and only if Ax = b has at most one solution for all b. To put it another way, the linear map associated to A is onto if and only if it is one-to-one. Again, be careful - this only applies to square matrices. Finally we talked about subspaces of R^n; these are subsets, containing the zero vector, that are closed under addition and scalar multiplication. We saw some examples: obvious ones like lines and planes through the origin, and then some less obvious ones such as the span of a set of vectors, and the column space and null space of a matrix.

Homework 3 is now posted on the Google calendar. Since Friday is a holiday, the homework will be due Thursday instead.

In order for you to be able to ask questions about this homework, I am changing my Thursday office hours to Wednesday from 1-2:30pm for this week only.

Study tip: use the worksheets from class to study for the quizzes. Worksheets are posted in the column on the left of this page.

As I'm sure you're aware, Friday 7/3 is a holiday, so there will be no class. There will be a quiz as usual on Wednesday.

Homework 3 will be due on Thursday 7/2 instead of Friday. I will post the problems sometime on Monday to give you a bit more time to work on them. If you're going miss class on Thursday, you can give me the homework in class on Wednesday or put it under my door anytime before 10am on Thursday.

What did we learn this week? Looking back it seems like we've covered a lot of ground so far. There was certainly a lot of definitions. Mainly we talked about linear systems, how to solve them, and several different ways of thinking about linear systems.

Day 1: Linear equations and linear systems, solution, solution set, equivalence of linear systems, consistency and inconsistency. Matrices, elementary row operations, row-equivalence of matrices. We saw that if two linear systems had row-equivalent augmented matrices, then the systems were equivalent, but not necessarily the other way around (eg by adding a row of zeros to the augmented matrix, the system is equivalent to the original one, but the matrices are different sizes so they can't be equivalent).

Day 2: Row-reduction and echelon forms of a matrix. We defined row echelon form (REF) and reduced row echelon form (RREF) of a matrix. Pivot positions and pivot columns. Row-reduction algorithm, forward and backward phases. Basic variables and free variables. We saw that the REF of the augmented matrix of a linear system is handy for determining whether the system is consistent, and if so whether or not is has a unique solution. The RREF is necessary to actually determine the solutions. We had an important theorem: A LS is consistent if and only if the rightmost column of the augmented matrix is not a pivot column; furthermore, if the system is consistent, then there is a unique solution if and only if there are no free variables, ie every column of the coefficient matrix is a pivot column.

Day 3: Vector and matrix equations. We defined vectors, addition and scalar multiplication of vectors. We defined linear combinations of vectors, the span of a set of vectors, and showed that the span of one non-zero vector is a line, while the span of two (independent) vectors is a plane. We showed that a linear system could be rewritten as a vector equation. We defined matrix multiplication, and showed that there is an equivalent matrix equation that describes the same linear system. We then asked, given a fixed matrix A, if the equation Ax=b is consistent for all vectors b. It turned out that the set of vectors b for which the equation is consistent is the span of the columns of A. Theorem 4 gave us several equivalent conditions for when A has the property that all vectors b are in the span of its columns.

Day 4: We started off with dot product, and then talked about why matrix multiplication is linear. We defined homogeneous and nonhomogeneous linear systems. Trivial and nontrivial solutions to a homogeneous equation. Particular solution. Then we moved on to discuss the structure of the set of solutions to a linear system. We saw that the solution set to a homogeneous equation is a line, plane or in general a span of a collection of vectors, so it always contain the origin. We saw how to describe the solution set in parametric vector form, ie how to write it as a span of vectors. Finally we saw that the solution set to a consistent nonhomogeneous equation is the solution set to the corresponding homogeneous system, translated by any particular solution of the nonhomogeneous equation. This was formalized by Theorem 6.

Day 5: We got more theoretical today. We defined linear independence, which formalized the notion of vectors "pointing in different directions." We defined linear dependence and independence of a set of vectors, dependence relations, and investigated some situations when we could easily determine when a set of vectors is dependent. We saw that the zero vector is the only vector that is dependent by itself; a set of two vectors is dependent if and only if one is a scalar multiple of the other; a set of vectors containing a dependent set is dependent. Finally, we said that a set of n vectors in m-dimensional space is automatically dependent if n is larger than m. Equivalently, the columns of an m by n matrix are automatically dependent when n is larger than m, ie when there are more columns than rows.

I announced in class today that my office hours will be changing. Unfortunately I announced the changes incorrectly. My office hours for the semester will be Mondays from 2:00 to 3:30, and Thursdays from 1:00 to 2:30.

As I said in class, you can access the homework by clicking on the link in the Friday 6/26 entry of the Google calendar on this page.

Remember that we have a quiz tomorrow! The worksheet from today is posted to the left, and that should give a good indication of what sort of questions the quiz will contain. Remember that definitions are also fair game.

The powers that be have decided to put us into 2 Evans for the entire class. We will no longer be in 3 Evans for the discussion section.

Here is the syllabus. You are responsible for reading it and understanding its contents.