## Weekly Homework Assignments

The weekly homework assignments will be published on this web page.

1. Homework: Due on Sept. 3rd in class: Problem 1, page 14, problem 4,
page 15, problem 3, page 25, problem 3 page 33.
2. Homework: Due on Sept. 10th in class: Problem 7 and problem 8,
page 33, problem 3 and 5, page 37, problem 5, page 48.

3. Homework. Due on Sept. 24th. 1. Prove that the sum of two subspaces of a vectorspace is a subspace.
Also do Problems 3,4,5,6 page 52.

4. Homework. Due on Oct. 1st : problems 1,3 page 68, problems 2,3 page 73, problem 8 page 74.

5. Homework. Due on Oct. 8th :1. Given a set S and a vector space V over a field F, show that
the functions from S to V can be given the structure of a vector space over F. When S={1,2,3}
and V=F^1 identify this vector space.
problems 4,6 page 87, problems 9,10 page 88.

6. Homework: Due on Oct. 29th: Do problems 4,5,7, 8,10, and 13 on pages 129-131.
7. Homework: Due Nov. 5th:
For the following problems let V be finite-dimensional real vector space and let B be a symmetric
bilinear form on V:
Show:
1) B is symmetric if and only if M_C(B) is symmetric, where M_C(B) is the Gram matrix of B with respect to C a basis of V.

2) B is nondegenerate if and only if M_C(B) is nonsingular.

3) If B is nondegenerate and W is a subspace of V then dim(W) + dim(W perp) = dim V, where W perp is the subspace
of vectors in V that are perpendicular to W.

4)
Determine a diagonal matrix D such that the real matrix A = [[2,1,0],[1,2,1],[0,1,2]] is congruent to D.

Do problems 2,3,4 on page 139,140.

8. Homework: Due Nov. 12th: Do problems 2,3,4, and 5 on page 149, 150.

9. Homework: Due Nov. 19th: Do problems 8,9,10, and 11 on page 176.
9. Homework: Due Dec. 6th: Do problems 5,6,7, and 8 on page 201.