These are by Ian Le, the Fall 2003 Math 55a Course Assistant

I'm going to try something different for the review session.  Here are
some review problems to think about.  I'd like you guys to work on these
and present solutions to each other in section.

Most of these are very simple.  I will write "{x}"  for tensor, "{+}" for
direct sum, "*" for dual, and "x" for Cartesian product.

Which of the following are true?
1) Hom(V,W) {x} Hom(V',W') = Hom(V{x}V',W{x}W')
2) Hom(V,W) {+} Hom(V',W') = Hom(V{+}V',W{+}W')
3) V**=V
4) V*=V
5) (U{+}V)*=U*{+}V*
6) (U{x}V)*=U*{x}V*
7) If e_1, e_2,...e_n is a basis for V, then (e_1+e_2)*=e_1*+e_2*.
8) Consider an mXn matrix.  The dimension of the vector space spanned by
the rows is the dimension of the vector space spanned by the columns.

Are the above isomorphisms canonical?

Some more questions:

If S, T are linear transformations of V and W, respectively, what is
det(S{+}T) in terms of det(S) and det(T)?  What is det(S{x}T)?  What is
det(S*)?

What are the eigenvalues and eigenvectors of S*? S{+}T? S{x}T?  What are
their minimal polynomials?  Characteristic polynomials?  Jordan forms?

What are det and trace in terms of the characteristic polynomial?

Let Z_n be the integers mod n.  How would you define Z_n {x} Z?  How about
Z_n {x} Z_m?  (Hint: use the universal definition of the tensor product)

Let V be the vector space of sequences of real numbers.  What is its dual
V*?  What is the cardinality of its basis?

Let e_1, e_2, ...e_n be a basis for V.  Let e_1* ... e_n* be the basis for
V* dual to this basis.  Show that e_1{x}e_1* + e_2{x}e_2* + ... e_n{x}e_n
is a unique element of V{x}V*, i.e. that it is independent of the basis we
choose (you might even say that it is a canonical element of V{x}V*).

Given an element of V{x}W, we can write it as v_1{x}w_1 + v_2{x}w_2 + ...
+ v_m{x}w{m} for some v_i, w_i in V and W.  What is the minimal m we need
to be able to write a given element of V{x}W?  (This is sometimes referred
to as the "rank" of an element in V{x}W.)