Math 225

IMPORTANT

• This homework consists of 5 questions of equal weight. Homework that does not satisfy the following requirements will lose points or will not be graded at all.

• You must use an A4-paper printout of this file and add extra pages as needed but only one side of each sheet must be used. Sheets must be attached together using a staple, paperclip or sheet protector.

• Homework must be submitted to your instructor before 3:30 pm of the due date. No Late homework will be accepted whatsoever. There is no make-up for homework.

• You must show all your work in well-organized English or mathematical sentences, and explain your reasoning carefully and in full.

• Similarities between homework papers that cannot be explained as coin- cidence will be treated as cheating.

Q1 Q2 Q3 Q4 Q5 Total

20 pt. 20 pt. 20 pt. 20 pt. 20 pt. 100 pt.

1

Question 1 (5+5+10=20 points) Let P2[x] be the space of polynomials whose degrees less than or equal to 2. If W = {p ∈ P2[x] | p(2) = 2p(1)}, then

(a) Show that W is a subspace of P2[x],

(b) Find a basis for W and determine its basis.

2

(c) Find a basis for W ⊥ where P2[x] is equipped with an inner product given by < p, q >=

∫ 1 0 p(x)q(x)dx.

3

Question 2 (20 points) If A ∈ M40×40(R) is such a matrix that A3 = 2I, show that B is invertible, where B = A2 − 2A + 2I.

4

Question 3 (20 points) Show that the set of all n × n upper triangular matrices W is a subspace of Mn×n(R). Find a basis for W and determine its dimension.

5

Question 4 (20 points) Prove that the eigenvalues of an upper triangular matrix A are the diagonal entries of A.

6

Question 5 (10+10 = 20 points) Let W1 and W2 be two subspaces of a finite dimensional vector space V.

(a) Prove that W1 ⊕ W2 = {α + β| α ∈ W1, β ∈ W2} and W1 ∩ W2 are subspaces of V.

7

(b) Show that dim(W1 ⊕ W2) = dim(W1) + dim(W2) − dim(W1 ∩ W2).

8