1. Let V be a vector space. Prove that if there is a spanning set
and if
is any linearly independent set of vectors in V, then m < n.
2. Prove that any two bases of a finite dimensional vector space have the same number of elements.
3. Let V be a vector space. Prove that if
are linearly independent vectors in V, and if
is a vector in V which is not in the space spanned by
then
are linearly independent.
4. Prove that in a finite dimensional vector space, any linearly independent set of vectors can be extended to a basis.
5. Prove that if T: V > W is a linear transformation, then
6. Let V be a vector space over F. Let T and U be linear transformations from V to V. Let
be a basis of V. Let X be the matrix representation of T with respect to B, and let Y be the martix representation of U with respect to B. Prove that YX is the matrix representation of UT with respect to B.
7. Let T: F ^{2} > F ^{2} be defined by T(x, y) = (x + y, x  y). Let A be the matrix representation of T with respect to the standard basis. and let B be the matrix representation of T with respect to the basis {(1, 2), (3, 4)}.
a) What is A?
b) What is B?
c) Find an invertible matrix P such that
8. Let X be an invertible nxn matrix. Define a function
by
Prove
a) 
f(A + B) = f(A) + f(B) 
for all nxn matrices A, B over F, 
b) 
f(cA) = c.f(A) 
for all nxn matrices A and all scalars c, 
c) 
f(AB) = f(A)f(B) 
for all nxn matrices A, B 
9. Define a relation on the set of nxn matrices over a field F by A ~ B if there exists an invertible nxn martix X such that
If A ~ B, then we say that A and B are similar. Prove that ~ is an equivalence relation. i.e., prove that
a) 
A ~ A 
for all nxn matrices A 
b) 
If A ~ B, then B ~ A 

c) 
If A ~ B and B ~ C, then A ~ C 
