1. If A is an invertible matrix and assuming exact arithmetic, are the solutions using Gaussian elimination and least squares identical? Prove your assertion 2. If  is an matrix,  and  is an  vector,...

1. If A is an invertible matrix and assuming exact arithmetic, are the solutions using Gaussian elimination and least squares identical? Prove your assertion

2. If
 is an
matrix,
 and
 is an
 vector, show that if

^T
 is orthogonal to the range of

3. If
 is an
full-rank matrix, show that

where

₁
and

_n
are the largest and smallest singular values of
 respectively.