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Let {₁,..., wk, Wk+1,...,n} be a basis of R" and c₁,...,CER be nonzero scalars. (a) Prove that S = {₁,..., nn} is a basis for R". (b) Suppose A € Mnxn (R) such that B = {wk+1,...,n} is a basis for Null(A). Prove that C={Aw₁,..., Awk) is a basis for Col(A).

Let {₁,..., wk, Wk+1,...,n} be a basis of R" and c₁,...,CER be nonzero scalars. (a) Prove that S = {₁,..., nn} is a basis for R". (b) Suppose A € Mnxn (R) such that B = {wk+1,...,n} is a basis for Null(A). Prove that C={Aw₁,..., Awk) is a basis for Col(A).

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 25CM: Find a basis B for R3 such that the matrix for the linear transformation T:R3R3,...
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Let {₁,..., wk, Wk+1,...,n} be a basis of R" and c₁,..., Cn ER be nonzero scalars. (a) Prove that S = {c₁w₁,..., nn} is a basis for R". (b) Suppose A € Mnxn (R) such that B = {wk+1,...,n} is a basis for Null(A). Prove that C = {Aw₁,..., Awk} is a basis for Col(A).
Transcribed Image Text:Let {₁,..., wk, Wk+1,...,n} be a basis of R" and c₁,..., Cn ER be nonzero scalars. (a) Prove that S = {c₁w₁,..., nn} is a basis for R". (b) Suppose A € Mnxn (R) such that B = {wk+1,...,n} is a basis for Null(A). Prove that C = {Aw₁,..., Awk} is a basis for Col(A).
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