Math 322

2nd Midterm

Spring 2001

Dr. Wilson

 

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

rank(T) + nullity(T) = dim V.

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

B = P-1AP.

8. Let X be an invertible nxn matrix. Define a function

f: Mn(F) -> Mn(F)

by

f(A) = X-1AX.

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

B = X-1AX.

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