MAT224 ASSIGNMENT 3
DUE BY FRIDAY JULY 24, 2020, 11:59 PM
Remark: 5 of the following questions will be marked, but you must do all of them. Each question that is marked is out of 5 for a total of 25 marks.
Question 1. Let U = span{(1,0,1,0),(1,1,1,0),(1,1,0,0)} and v = (2,0,−1,3). Find the vector in U closest to v. Question 2. If U is a subspace of Rn, show that U⊥⊥ = U.
Question 3. Let P be an orthogonal matrix.
(a) Prove that detP = 1 or detP = −1.
(b)IfdetP =−1,showthatI+P hasnoinverse. Hint: PT(I+P)=(I+P)T.
Question 4. Let A be a positive definite matrix. If a ∈ R, prove that aA is positive definite if and only if a > 0. a b
Question 5. (a) Show that a real 2×2 matrix is normal if and only if it is either symmetric or has the from −b a .
(b) Let U be a unitary matrix. Prove that |λ| = 1 for every eigenvalue λ of U (where |λ| is the norm of the complex number λ).
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