4.4 Change of Basis

1. Given the basis and coordinates, write the vector in terms of the standard basis. a. $\mathcal{B}=\left\{\left[\begin{array}{c}1\\ -3\end{array}\right],\left[\begin{array}{l}2\\ 1\end{array}\right]\right\},{\mathbf{x}}_{\mathcal{B}}={\left[\begin{array}{c}2\\ -2\end{array}\right]}_{\mathcal{B}}$
   $\left[\begin{array}{c}8\\ 2\end{array}\right]$
   b. $\mathcal{B}=\left\{\left[\begin{array}{l}1\\ 0\\ 2\end{array}\right],\left[\begin{array}{c}3\\ -1\\ 3\end{array}\right],\left[\begin{array}{l}0\\ 2\\ 2\end{array}\right]\right\},{\mathbf{x}}_{\mathcal{B}}={\left[\begin{array}{c}-2\\ 1\\ 3\end{array}\right]}_{\mathcal{B}}$
   $\left[\begin{array}{c}4\\ 2\\ 8\end{array}\right]$
2. Given the vector $\mathbf{x}$ written in terms of the standard basis, find the coordinate vector of $\mathbf{x}$ with respect to $\mathcal{B}$.
   a. $\mathbf{x}=\left[\begin{array}{l}-2\\ -8\end{array}\right],\mathcal{B}=\left\{\left[\begin{array}{c}1\\ -3\end{array}\right],\left[\begin{array}{l}2\\ 1\end{array}\right]\right\}$
   $\left[\begin{array}{c}-18\\ -2\end{array}\right]$
   b. $\mathbf{x}=\left[\begin{array}{c}2\\ -1\end{array}\right],\mathcal{B}=\left\{\left[\begin{array}{l}4\\ 3\end{array}\right],\left[\begin{array}{l}5\\ 4\end{array}\right]\right\}$
   $\left[\begin{array}{c}3\\ 2\end{array}\right]$
   c. $\mathbf{x}=\left[\begin{array}{c}1\\ 3\\ -2\end{array}\right],\mathcal{B}=\left\{\left[\begin{array}{l}0\\ 3\\ 2\end{array}\right],\left[\begin{array}{l}1\\ 3\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ -3\end{array}\right]\right\}$
   $\left[\begin{array}{c}1\\ 12\\ -4\end{array}\right]$
3. Use $\mathbf{x}=\left[\begin{array}{c}1\\ 3\\ -2\end{array}\right],\mathcal{B}=\left\{\left[\begin{array}{l}0\\ 3\\ 2\end{array}\right],\left[\begin{array}{l}1\\ 3\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ -3\end{array}\right]\right\}$ to compute $[A\mathbf{x}{]}_{\mathcal{B}}$ and $A_{\mathcal{B}}$ where $A=\left[\begin{array}{ccc}2& 0& 0\\ 9& -1& 3\\ 6& -2& 4\end{array}\right]$.
   $A_{\beta }=\left[\begin{array}{ccc}15& -3& 7\\ 33& -3& 9\\ -14& 6& -12\end{array}\right]$
   $[Ax{]}_{\beta }\left[\begin{array}{c}-8\\ 6\\ 28\end{array}\right]$
4. Find the change of basis matrix for the given bases.

$${\mathcal{B}}_1=\left\{\left[\begin{array}{c}1\\ -3\end{array}\right],\left[\begin{array}{l}2\\ 1\end{array}\right]\right\},{\mathcal{B}}_2=\left\{\left[\begin{array}{l}4\\ 3\end{array}\right],\left[\begin{array}{l}5\\ 4\end{array}\right]\right\}$$