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 thatC = {Aw₁,..., Awk} is a basis for Col(A).

Note that we used one fact in this prove which state that if B is basis of V and C is any linearly…

Answered: Let {₁,..., wk, Wk+1,...,n} be a basis…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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).