18.06 Problem Set 6

Due Wednesday, May 1

Problem 1. Find a diagonalization for A by an orthogonal matrix Q where

A = [ \begin{matrix}0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{matrix} ] .

Problem 2. Find the singular value decomposition for the matrix

B = [ \begin{matrix}1 & −1 \\ 2 & −6 \end{matrix} ] .

Problem 3. Suppose L : ℝ^{4} ⟶ ℝ^{2} is a linear transformation with L(1, 0, 0, 1) = (2, 3) and L(0, 1, 1, 0) = (1, 5). Find L(6, 2, 2, 6). Problem 4. Let L : ℝ^{2} ⟶ ℝ^{3} by

L ([ \begin{matrix}x \\ y \end{matrix} ]) = [ \begin{matrix}x − 2y \\ 2x + y \\ x + y \end{matrix} ] .

Let S, T be the standard bases of ℝ^{2}, ℝ^{3} respectively, and S′ = {(1, −1), (0, 1)}, T′ = {(1, 1, 0), (0, 1, 1), (1, −1, 1). Find the matrix for the linear transformation L with respect to

(a) S and T.

(b) S′ and T′.

(c) S and T′.

(d) S′ and T. Problem 5. Let M_{2} be the (four dimensional) vector space of 2 × 2 matrices with real number entries. Define a linear transformation L : M_{2} ⟶ M_{2} by L(A) = A^{T}. Let

S = \{[ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]\}

and

T = \{[ \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ]\}

Find the matrix for L with respect to

(a) S.

(b) T.

(c) S and T.

(d) T and S.