Question 106640
*[Tex \LARGE T:R^n\rightarrow R^m] such that *[Tex \LARGE T(\v{x})=A\v{x}]


Remember *[Tex \LARGE T(\v{x})=\v{b}], so this means *[Tex \LARGE A\v{x}=\v{b}]



Now plug in *[Tex \LARGE A=\begin{bmatrix} 1 & -3 &  2 \\ 0 & 1 & -4 \\ 3 & -5 & -9 \end{bmatrix}] and *[Tex \LARGE \v{b}=\begin{bmatrix} 6\\ -7 \\ -9 \end{bmatrix}]



*[Tex \LARGE \begin{bmatrix} 1 & -3 &  2 \\ 0 & 1 & -4 \\ 3 & -5 & -9 \end{bmatrix}\v{x}=\begin{bmatrix} 6\\ -7 \\ -9 \end{bmatrix}]


So which vectors *[Tex \Large \v{x}] will satisfy this matrix vector equation?


To find out, simply set up the augmented matrix *[Tex \LARGE \left[\textrm{A \v{b}}\right]]


*[Tex \LARGE \begin{bmatrix} 1 & -3 &  2 & 6\\ 0 & 1 & -4 & -7\\ 3 & -5 & -9 & -9\end{bmatrix}]


Now row reduce the augmented matrix to get


*[Tex \LARGE \begin{bmatrix} 1 & 0 &  0 & -5\\ 0 & 1 & 0 & -3\\ 0 & 0 & 1 & 1 \end{bmatrix}]



So *[Tex \Large \v{x}=\begin{bmatrix} -5\\ -3\\ 1 \end{bmatrix}] which means *[Tex \Large \v{x}] is unique