quiz3 - Name: Linear Algebra Quiz 3 June 13, 2006 1. 2 1 3...

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Unformatted text preview: Name: Linear Algebra Quiz 3 June 13, 2006 1. 2 1 3 2 , } and γ = { , }. 1 1 1 1 (a)(4 points) Find the transition matrices from β -coordinates to standard coordinates, γ coordinates to standard coordinates, and γ -coordinates to β -coordinates, i.e., find [id]STD , β [id]STD , and [id]γ . γ β x 2x + y (b) (4 points) Let L : R2 → R2 by L = . Show that L is a linear transfory −x + 3y mation and find a matrix A such that L(x) = Ax for all x ∈ R2 . (c) (4 points) Find the matrix representing L with respect to the bases β to β and β to γ , i.e., find [L]β and [L]γ . β β 3 2 (d) (4 points) Suppose x = 2 − . Find the coordinate representation of x with 1 1 respect to the ordered bases β , γ , and the standard basis, i.e., find [x]β , [x]γ , and [x]STD . (e) (4 points) Find the coordinate representation of L(x) with respect to the ordered bases β , γ , and the standard basis, i.e., find [L(x)]β , [L(x)]γ , and [L(x)]STD . Consider R2 with ordered bases β = { Solution: (a) [id]STD = β 0 −1 . 12 x (b) L = y 21 , [id]STD = γ 11 21 −1 3 x y 32 , [id]γ = [id]γ [id]STD = β STD β 11 1 −2 −1 3 21 11 = so L is left multiplication by a matrix and is hence linear. 21 1 −1 so [L]β = [id]β [L]STD [id]STD = STD β STD β −1 3 −1 2 41 1 −2 21 2 . Similarly, [L]γ = [id]γ [L]STD [id]STD = STD β STD β −3 1 −1 3 −1 3 1 0 −1 2 1 2 3 (d) [x]β = , [x]STD = , [x]γ = [id]γ [x]β = = β 12 −1 −1 −1 1 41 2 7 (e) [L(x)]β = [L]β [x]β = = , [L(x)]γ = [id]γ [L(x)]β = β β −3 1 −1 −7 21 7 7 7 (weird!), [L(x)]STD = [id]STD [L(x)]β = = β −7 11 −7 0 (c) (b) says [L]STD = STD 1 21 21 = −1 3 11 1 3 −1 = 1 −2 3 . 0 −1 12 7 −7 = ...
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.

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