1. WORKING WITH BASIC DEFINITIONS
   T is a linear transformation from X to Y.
   Prove that if {x1, ..., xn} is a basis for X,
   then {Tx1, ..., Txn} is a spanning set for the image of T.
   (That's the set of all vectors of the form Tx; it's a subspace of Y.)

   Is the set {Tx1, ..., Txn} necessarily a *basis* for the image of T?
   Is the (original) statement still true if {x1, ..., xn} is a spanning set,
     but not necessarily a basis, for X?

2. FUNDAMENTAL SUBSPACES OF A MATRIX
   A is given below. Find bases for NS(A), RS(A), and CS(A). Find rk(A).
   (Only use one round of elimination!)
[ 0  2 -3 1 2 ]
[ 0 -2  3 3 1 ]
[ 0  4 -6 6 7 ]

3. BASES FOR THE SPAN OF A COLLECTION OF VECTORS
   (a) Find a basis for the subspace spanned by the following vectors:
       <-2, 4, 1, 2>,  <4, 2, 3, -1>,  <2, 6, 4, 1>
   (b) Find an *orthonormal* basis for the span of these vectors:
       <1, 1, 1>,  <1, 2, 2>,  <1, 0, 1>
   (c) Find a basis for the subspace of P_2 spanned by these polynomials:
       2 - 3x + 4x^2 - 5x^3,  4 - x + 15x^2 - 2x^3,  2 + 2x + 11x^2 + 3x^3
   (d) Find an *orthonormal* basis for the span of these polynomials.
       Use the inner product from C[-1,1]:
       i.e., f.g = integral of f(x)g(x)dx from -1 to 1.
       1 - x, 1 + x, 1 + x + x^2

4. CHANGE OF BASIS AND TRANSITION MATRICES
   (a) Find the transition matrix from B to C, and [x]_C:
       (1) B = {<1, 1>,  <1, 0>}, C = {<2, 3>,  <4, 2>}, [x]_B = <-1, 2>
       (2) B={1+x+x^2,1+x,x+x^2}, C = standard basis for P_2, [x]_B=<2,3,-2>.
   (b) B = {<1, 1>,  <1, -1>} and C is another basis for R^2. x = <4, 2>.
       You are given the transition matrix from B to C (below). Find
       (1) [x]_B and [x]_C
       (2) The transition matrix from C to B
       (3) The vectors {v1, v2} that form C
(You will get fractions, but "mild" ones.)
P = [ 0.6 -0.2 ]
    [ 0.2  0.6 ]

5. LINEAR TRANSFORMATIONS
   Find the domain and range (as Hill defines it) of the given function T.
   Determine whether T is a linear transformation.
   If it is, find the induced matrix A=[T].
   (a) T(x, y) = (2x, y)
   (b) T(x, y, z) = (xy,zx)
   (c) T(a0 + a1 x + a2 x^2 + a3 x^3) = a1 + 2 a2 x + 3 a3 x^2
   (d) T(x, y, z) is "the reflection of (x,y,z) through the xy-plane,
       rotated by 30 degrees around the z-axis."
       (In other words, T is the transformation which
        rotates the xy-plane by 30 degrees and reflects the z-axis.)

6. LEAST SQUARES
   Do exercise 14 of Hill section 4.3. Then find the best-fit
   *quadratic* equation.
   Finally, do exercise 20---we haven't talked about perpendicular
   projection, but it sounds like you will be expected to know it!

7. DETERMINANTS
   Evaluate this determinant:
| -1   1   0   1   2   1  0 |
|  4  -4   0   2   1  -1  0 |
|  1   2   1   1   1   1  1 |
|  2   8   0   0   0   0  0 |
|  0  -3   0   0   0   0  0 |
|  1   5   0   0   0   1  0 |
|  0  -7   1   2  -2   2  0 |
   Also find the trace of the above matrix.
   Also evaluate the determinant of
[1 1] [4 1]^-1 [1 3] [2 7]
[1 1] [5 6]    [2 4] [3 6]

8. EIGENVALUE PROBLEMS
   Find the eigenvalues of A (below), and bases for its eigenspaces.
   Do the same for the matrix I-A, without doing extra work.
   Decide whether A is invertible, also without doing extra work.
    [-6  12  -1 ]
A = [-6  11   0 ]
    [ 0   0   2 ]

9. DIAGONALIZABILITY
   Is the matrix A from #8 diagonalizable?
   Diagonalize the 2x2 matrix [2, 0 ; 1, 1] and compute its 12th power.
   What do you know about the problem of diagonalizing this matrix,
   without doing any work? (Don't try to calculate it.)
   [ 0 2 0 1 ]
   [ 2 0 3 0 ]
   [ 0 3 0 7 ]
   [ 1 0 7 8 ]

10. PROOFS
   Let A be a matrix satisfying A^2-5A-6=0. If lambda is an eigenvalue,
   prove lambda=6 or lambda=-1.
   If A is an mxn matrix, and x is a vector in R^n, prove (A^T A x) . x
   is ALWAYS >= 0. If A is invertible, prove it's only 0 if x=0.