In the vector space $\complex{3}$, compute the vector representation $\vectrep{B}{\vect{v}}$ for the basis $B$ and vector $\vect{v}$ below. \begin{align*} B&=\set{ \colvector{2\\-2\\2},\, \colvector{1\\3\\1},\, \colvector{3\\5\\2} } & \vect{v}&=\colvector{11\\5\\8} \end{align*}

Rework Example CM32 replacing the basis $B$ by the basis \begin{equation*} C= \set{ \begin{bmatrix} -14 & -9 \\ 10 & 10 \\ -6 & -2 \end{bmatrix},\, \begin{bmatrix} -7 & -4 \\ 5 & 5 \\ -3 & -1 \end{bmatrix},\, \begin{bmatrix} -3 & -1 \\ 0 & -2 \\ 1 & 1 \end{bmatrix},\, \begin{bmatrix} -7 & -4 \\ 3 & 2 \\ -1 & 0 \end{bmatrix},\, \begin{bmatrix} 4 & 2 \\ -3 & -3 \\ 2 & 1 \end{bmatrix},\, \begin{bmatrix} 0 & 0 \\ -1 & -2 \\ 1 & 1 \end{bmatrix} } \end{equation*}

Prove that the set $S$ below is a basis for the vector space of $2\times 2$ matrices, $M_{22}$. Do this by choosing a natural basis for $M_{22}$ and coordinatizing the elements of $S$ with respect to this basis. Examine the resulting set of column vectors from $\complex{4}$ and apply the Coordinatization Principle. \begin{equation*} S=\set{ \begin{bmatrix} 33 &99\\ 78 & -9 \end{bmatrix},\, \begin{bmatrix} -16 & -47\\ -36 & 2 \end{bmatrix},\, \begin{bmatrix} 10 & 27\\ 17 & 3 \end{bmatrix},\, \begin{bmatrix} -2 & -7\\ -6 & 4 \end{bmatrix} } \end{equation*}

The set $B=\set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3,\,\vect{v}_4}$ is a basis of the vector space $P_3$, polynomials with degree 3 or less. Therefore $\vectrepname{B}$ is a linear transformation, according to Theorem VRLT. Find a “formula” for $\vectrepname{B}$. In other words, find an expression for $\vectrep{B}{a+bx+cx^2+dx^3}$. \begin{align*} \vect{v}_1&=1 - 5x - 22x^2 + 3x^3 & \vect{v}_2&=-2 + 11x + 49x^2 - 8x^3\\ \vect{v}_3&=-1 + 7x + 33x^2 - 8x^3 & \vect{v}_4&=-1 + 4x + 16x^2 + x^3 \end{align*}