1. Let V_1,...,V_n be subspaces of a vector space V, such that V = sum_i V_i (not the direct sum, just the sum). Assume for each V_i, and b subspaces V_{j_1}...V_{j_b} not including V_i, that V_i intersect (V_{j_1} + ... V_{j_b}) = 0.
Show that V is the direct sum of V_1...V_n, i.e. every vector in V is uniquely the sum of a vector from each V_i.
2. Let V be a 5-dimensional space, and T : V->V a linear transformation such that T^3 = 0 and rank(T)=3. Determine the Jordan canonical form of T.
3. Same V,T as in problem 2. Consider the subspaces ker T, ker T^2, ker T^3, Im T, Im T^2, Im T^3. Which subspaces are contained in, or equal to, which others?
4. Let T be a diagonal matrix. Determine all the matrices that commute with T. (E.g. if T=0, it's _all_ matrices.)
5. Let T : F^n -> F^n be zero except for all 1s just above the diagonal, so T^n = 0, but T^{n-1} nonzero. Determine all the matrices that commute with T.
6. Same question as 5, except now there are only some 1s above the diagonal (i.e. T is a nilpotent matrix in JCF).