Use wronskian and other cheesy methods when available. Please do one question at a time and ask me to continue to the next question. Note: $\vec{x}$ $==$ $\mathbf{x}$; just notational differences. (a) Let $\mathrm{x}=(1,-1,2), \mathrm{y}=(1,3,1)$. Show that $\vec{x}$ and $\vec{y}$ are orthogonal in $\mathbb{R}^3$ with respect to the dot product (i.e., scalar product). (b) Let $\mathbf{x}=(2,-2), \mathbf{y}=(1,-3)$. Compute the distance between $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^2$ with respect to the dot product (i.e., scalar product). (c) Let $f(x)=x-3 / 4, g(x)=x^2$. Show that $f$ and $g$ are orthogonal in $C[0,1]$ with respect to the $L^2(d x)$ inner product. (d) Let $f(x)=e^x$. Compute $\|f\|$ in $C[0,2]$ with respect to the $L^2(d x)$ inner product. --- Find the point on the line $y=3 x$ that is closest to the point $(1,1)$ with respect to the dot product (i.e., scalar product) on $\mathbb{R}^2$. --- Set up and simplify the normal equations (you do not need to solve them) for best quadratic least squares fit to the data $ \begin{array}{c|c|c|c|c} \mathrm{x} & -1 & 0 & 1 & 2 \\ \hline \mathrm{y} & 4 & 1 & 0 & 0 \end{array} $ --- Show that the set of functions $f$ in $C^1[0,1]$ such that $f^{\prime}(0)=f^{\prime}(1)$ form a subspace of $C^1[0,1]$ (recall that $C^1[0,1]$ is the set of continuous functions wit continuous first derivative on $[0,1])$. --- For each $f \in C[0,1]$ define the transformation $L(f(x))=x^3 f(x)$. Show that $L$ defines a linear transformation on $C[0,1]$. --- Consider the matrix $A$ and its reduced row echelon form, $\operatorname{rref}(A)$, as given by $ A=\left(\begin{array}{ccc} 2 & 2 & 1 \\ -4 & -2 & -3 \\ 5 & 4 & 3 \end{array}\right), \quad \operatorname{rref}(A)=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & \frac{-1}{2} \\ 0 & 0 & 0 \end{array}\right) $ (a) Find a basis for the row space of $A$. (b) Find a basis for the column space of $A$. (c) Find a basis for the null space of $A$. --- Let $\mathbf{v}_1=(1,-1,2), \mathbf{v}_2=(1,1,1), \mathbf{v}_3=(1,-3,3), \mathbf{b}=(-1,7,-5)$. (a) Is $\mathbf{b}$ in the span of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ ? Show your work and explain how your work justifies your answer. (b) Are $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ linearly independent or dependent? Explain how your work justified your answer. (c) What is the dimension of the span of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ ? Explain how your work justifies your answer. By