- #1

StandardBasis

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## Homework Statement

Hi there! First time user, so I hope I do this right. The question is: Let A be an 8x5 matrix of rank 3, and let b be a nonzero vector in N(A

^{T}). First, prove that the system Ax=b must be inconsistent. Then, how many least squares solutions will the system have?

## Homework Equations

## The Attempt at a Solution

I got the first part fine (I think...). I assume that Ax=b is consistent, and look for a contradiction. There is one: for Ax=b to be consistent, b must be in the column space of A. But we are told b is in the nullspace of A

^{T}, which is the orthogonal complement of R(A). So b isn't actually in R(A), and thus Ax=b is inconsistent by the fact that b isn't a linear combination of A's columns.

Now for the second part, I know I need to find all solutions to A

^{T}Ax=A

^{T}b. In class we proved that if A is mxn with rank n, then there is a unique solution to the normal equations. Obviously rank is not n here, so I'm not sure what to do. I'm used to systems having either 0, 1, or infinitely many solutions... so I'm tempted to say infinitely many? But I can't justify that.