Chapter 5.4, Problem 44E

True or false? In Exercises $43$ and $44$, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.

$\left(\text{a}\right)$ If $A$ is an $m\times n$ matrix, then $R\left(A\right)$ and $N\left(A^T\right)$ are orthogonal subspaces of $R^n$.

$\left(\text{b}\right)$ The set of all vectors orthogonal to every vector in a subspace $S$ is the orthogonal complement of $S$.

$\left(\text{c}\right)$ Given an $m\times n$ matrix $A$ and a vector $b$ in $R^m$, the least square problem is to find $x$ in $R^n$ such that ${\Vert Ax\text{-}b\Vert }^2$ is minimized.