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let W = span(ē₁,ē2). Let T : R² → W be defined by ™ ( [23]) = -8] What is dim(W)? What is ker(T)? and what is ran(T)? Is T an isomorphism? (Recall that T is an isomorphism if and only if ker(T) = {0} and ran(T) = W.) Find a matrix A Є R2×2 so that [T(F)] = Ar

let W = span(ē₁,ē2). Let T : R² → W be defined by ™ ( [23]) = -8] What is dim(W)? What is ker(T)? and what is ran(T)? Is T an isomorphism? (Recall that T is an isomorphism if and only if ker(T) = {0} and ran(T) = W.) Find a matrix A Є R2×2 so that [T(F)] = Ar

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 16CM
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In R³, let e1 = and e2 = = and let W = span(e1, №2). Let T: R² → W be defined by (a) What is dim(W)? T (E)) = (b) What is ker(T)? and what is ran(T)? x (c) Is T an isomorphism? (Recall that T is an isomorphism if and only if ker(T) = {0} and an(T) = W.) ran (d) Find a matrix A € R2×2 so that [T(x)] = Ax 33 where B = (1, 2) is the natural basis for W
Transcribed Image Text:In R³, let e1 = and e2 = = and let W = span(e1, №2). Let T: R² → W be defined by (a) What is dim(W)? T (E)) = (b) What is ker(T)? and what is ran(T)? x (c) Is T an isomorphism? (Recall that T is an isomorphism if and only if ker(T) = {0} and an(T) = W.) ran (d) Find a matrix A € R2×2 so that [T(x)] = Ax 33 where B = (1, 2) is the natural basis for W
Note: In a certain sense, made precise by this problem R² is kind-of a subspace of R³. It is not literally a subspace, but R2 can be very naturally identified with W, some of you noticed this similarity at the very beginning when I asked questions about the relationship between R² and R3, though this problem makes it very precise. In a similar fashion, Rn is "kind-of" a subspace of Rk whenever k ≥ n.
Transcribed Image Text:Note: In a certain sense, made precise by this problem R² is kind-of a subspace of R³. It is not literally a subspace, but R2 can be very naturally identified with W, some of you noticed this similarity at the very beginning when I asked questions about the relationship between R² and R3, though this problem makes it very precise. In a similar fashion, Rn is "kind-of" a subspace of Rk whenever k ≥ n.
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