
# 17.2: Review Problems

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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.