Let A = _ 1 1 0 0 0 0 1 1 _ . a. Given any x R4, find u R(A) and v N(A) so that x = u +

QUESTION:

Let \(A=\left[\begin{array}{rrrr}1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right]\).

a. Given any \(\mathbf{x} \in \mathbb{R}^4\), find \(\mathbf{u} \in \mathbf{R}(A)\) and \(\mathbf{v} \in \mathbf{N}(A)\) so that \(\mathbf{x}=\mathbf{u}+\mathbf{v}\).

b. Given \(\mathbf{b} \in \mathbb{R}^2\), give the unique element \(\mathbf{x} \in \mathbf{R}(A)\) so that \(A \mathbf{x}=\mathbf{b}\).