$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 17.2: Review Problems

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

1. Let $$L:U \Rightarrow V$$ be a linear transformation. Suppose $$v \in L(U)$$ and you have found a vector $$u_{ps}$$ that obeys $$L(u_{ps})$$ = v.

Explain why you need to compute $$ker L$$ to describe the solution set of the linear system $$L(u) = v$$.

2. Suppose that $$M$$ is an $$m \times n$$ matrix with trivial kernel. Show that for any vectors $$u$$ and $$v$$ in $$\mathbb{R}^{m}$$.

a) $$u^{T}M^{T}Mv = v^{T}M^{T}Mu$$.

b) $$v^{T}M^{T}Mv \geq 0$$. In case you are concerned (you don't need to be) and for future reference, the notation $$v \geq 0$$ means each component $$v^{i} \geq 0$$.

($$\textit{Hint:}$$ Think about the dot product in $$\mathbb{R}^{n}$$.)

3. Rewrite the Gram-Schmidt algorithm in terms of projection matrices.

4. Show that if $$v_{1}, \cdots , V_{k}$$ are linearly independent that the matrix $$M = (v_{1} \cdots v_{k})$$ is not necessarily invertible but the matrix $$M^{T}M$$ is invertible.