Question

In Exercises 25 through 30 , find the matrix B of the linear transformation $\mathit{T}\mathbf{\left(}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{A}\overrightarrow{\mathbf{x}}$ with respect to the basis$\mathfrak{J}\mathbf{=}\left({\overrightarrow{v}}_1,\dots ..,{\overrightarrow{v}}_m\right)$.

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}-1& 1& 0\\ 0& -2& 2\\ 3& -9& 6\end{array}\right]\mathbf{;}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{=}\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\mathbf{;}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{=}\left[\left.\begin{array}{c}1\\ 2\\ 3\end{array}\right];{\overrightarrow{v}}_3=\left[\begin{array}{c}1\\ 3\\ 6\end{array}\right.\right]$

Short answer

The matrix is, $B=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right]$.

Step-by-step solution

Consider the vectors.

The matrix and vectors are,

$A=\left[\begin{array}{ccc}-1& 1& 0\\ 0& -2& 2\\ 3& -9& 6\end{array}\right];{\overrightarrow{\mathrm{v}}}_1=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right];{\overrightarrow{\mathrm{v}}}_2=[\begin{array}{c}1\\ 2\\ 3\end{array}];{\overrightarrow{\mathrm{v}}}_3=[\begin{array}{c}1\\ 3\\ 6\end{array}]$

Compute the matrix using formula.

The formula is, $B=S^{-1}AS$.

Compute the matrix and the inverse of S.

$$\begin{gathered} S=\left({\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3\right) \\ =\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right] \\ \Rightarrow S^{-1}=\frac{1}{1}\left[\begin{array}{ccc}3& -3& 1\\ -3& 5& -2\\ 1& -2& 1\end{array}\right] \end{gathered}$$

Substitute these values in the formula.

$$\begin{gathered} B=S^{-1}AS \\ \Rightarrow B=\frac{1}{1}\left[\begin{array}{ccc}3& -3& 1\\ -3& 5& -2\\ 1& -2& 1\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -2& 2\\ 3& -9& 6\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right] \\ \therefore B=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right] \end{gathered}$$

Final answer.

The matrix is, $B=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right]$.