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
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
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
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.