## Spring 2001

### Dr. Wilson

1/29 p5: 5

1/31 Prove that if F is a field then FxF is a vector space. p5: 2; pp10-11: 2, 3; p15: 1

2/5 p21: 1, 2, 5

2/7 pp26-27: 1, 3, 5, 8, 9; Prove Example 3 on p30; pp33-34; 3, 5; p39: 1.

2/12 No homework. We spent the whole period going over the homework from the class before.

2/14 p48: 1, 2, 4.

2/21 Find the space spanned by the vectors (1, 2, 3), (4, 5 6), (7, 8, 9). Compute the dimension by finding a basis for the space. Hint: Row reduce the matrix whose rows are these vectors.

3/12 1. Prove that the null space of a linear transformation is a subspace of the domain.

2. Prove that if T: V -> W is a linear transformation, the range of T is a subspace of W.

p73: 1, 4, 5, 7, 8

3/14 Prove that if A is an mxn matrix and B is a nxp matrix over the same field that

(AB)T = BTAT

3/19 Prove that the composition of two isomorphism is an isomorphism; p86: 6

3/21 p95: 1, 4, 6

3/26 p54: 1; pp95-96: 5, 9

4/2 Prove all of the ring axioms for the set of polynomials over a field.

4/4 1. Let A be an element of an algebra over a field F, and let f(x), g(x) be polynomials with coefficients in F. Prove that

(f + g) (A) = f(A) + g(A)

(fg)(A) = f(A)g(A)

2. Let R be a commutative ring with 1. Let a and b be elements of R. Show that

{ra + sb| r and s range over all elements of R}

is an ideal.

pp122-123: 1, 2, 4;p134: 1,4

4/18 Let A be an element of an algebra over a field F. Show that the set of all polynomials which have A as a root

{f(x) | f(A) = 0 }

is an ideal of F[x].

p134: 2.

4/25 p139: 3; pp148-149: 3, 5; p155: 1, 2, 7

5/2 p148: 2; p155: 3, 4: pp162-163: 1, 4, 7, 10

5/7 1. Show that if A is an upper triangular matrix, det A is the product of the main diagonal entries.

2. Show that if I is an ideal of F[x] and T is a linear operator on the vector space V that

is a T-invariant subspace.

3. Show that if

that the annihilator

is an ideal.