# Midterm - N A ≥ m 2 or dim N B ≥ m 2 Problem#4(30 pts a...

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Math 104 : Midterm Instructions: Complete the following 4 problems. Remember to show all your work. No notes or calculators are allowed. Please sign below to indicate you accept the honor code. Name: SUID: Signature: Problem 1 2 3 4 Total Score

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Problem #1. (20 pts) Let w 1 , w 2 and w 3 be three vectors in C 3 . Let v 1 = w 1 - w 3 , v 2 = w 1 + w 2 , v 3 = w 1 + λ w 3 , and v 4 = 2 w 1 + w 2 - w 3 . Where here λ C . For what value λ 0 is it always true that when λ = λ 0 , v 1 , v 2 , v 3 and v 4 never span C 3 . Justify your answer. (Hint: Rewrite the prob- lem using matrices).
Problem #2. (30 pts) Let v = 2 sin θ - 2 cos θ R 2 Let A R 2 × 2 denote the matrix which gives orthogonal projection onto span ( v ). a) Determine A . b) Determine N ( A ) and R ( A ). c) Determine a full QR factorization of A .

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Problem #3. (20 pts) Let A, B C m × m suppose that AB = 0 and BA = 0. a) What, if any, is the relationship between the null space of A and the column space of B ? Justify your answer. b) Show that either dim

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Unformatted text preview: N ( A ) ≥ m 2 or dim N ( B ) ≥ m 2 . Problem #4. (30 pts) a) Suppose that A,B ∈ C m × m are unitary matrices. Verify that A * and AB are also unitary. (Hint: Use the algebraic properties of the adjoint) b) Let v 1 = 1 5 3-4 , v 2 = 1 5 4 3 , v 3 = 1 ∈ R 3 Verify that { v 1 , v 2 , v 3 } is an orthonormal basis of R 3 . Justify your answer. c) Let w 1 = √ 2 2 1 1 , w 2 = -1 , w 3 = √ 2 2 -1 1 ∈ R 3 be a set of orthonormal vectors. Determine the orthogonal matrix U ∈ R 3 × 3 so that U v i = w i for i = 1 , 2 , 3 here the v i are given in b). (Hint: Use part a) )...
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