Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.3: Change Of Basis
Problem 22EQ
See similar textbooks
icon
Related questions
Question
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).
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS