﻿ 1 Consider the least squares problem mln. Aac b where A is m x n and rankA is S n. Let denote t
X

### Question

(1) Consider the least squares problem mln. Aac b where A is m x n and rank(A) is S n. Let denote the shortest solution to the above least squares problem. Explain why is always in the row space of A

### Solution

xx xxxxx from xxx fact xxxx xxx xxxxxxxx and xxx nullspace xxx xxxxxxxxxx xxxxxxxxxxx. This xxxxxxxxxxy implies xxxx xxx x real x×xx×x matrix xx that

If yxx require x xxxxx xx the xxxx that xxx xxxxxxxxx xxx rowspace xxx orthogonal xxxxxxxxxxx, xxxxxy xxxx every xxxxxxxx linear xxxxxxx xxxx xxxxxx contain xxxx a xxxxx.

xx xxxxx xxxx xxx fact xxxx the xxxxxxxx xxx xxx nullspace xxx xxxxxxxxxx complements. xxxx xxxxxxxxxxy xxxxxxx that xxx a xxxx x×xx×x xxxxxx AA that

xn=ker(x)xxx(Ax)Rn=ker(A)col(AT)
so xxxx any xxnxxx xxx xx written xxxxxxxy as x=xr+xxx=xx+xx xxxxx xxxxx(x)xxxxx(x) xxx xxxxx(xx)xrcol(AT).

xx yxx require x proof xx xxx xxxx that xxx nullspace xxx xxxxxxxx xxx xxxxxxxxxx complements, xxxxxy much xxxxy xxxxxxxx xxxxxx algebra text xxxxxx contain xxxx x xxxxx.