Let A = \(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\) and define T:R2 → R2 by T(x) = Ax, find the image under T of

\(u = \begin{bmatrix}1\\ -3\end{bmatrix}\) and \(v = \begin{bmatrix}1\\ 5\end{bmatrix}\)

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Solution:

Given

A = \(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\)

\(u = \begin{bmatrix}1\\ -3\end{bmatrix}\)

\(v = \begin{bmatrix}1\\ 5\end{bmatrix}\)

Now, We have to find image of T under u and v

T(u) = Au = \(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\) \(\begin{bmatrix}1\\ -3\end{bmatrix}\)

= \(\begin{bmatrix}-3\\ -1\end{bmatrix}\)

T(v) = Av = \(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\) \(\begin{bmatrix}1\\ 5\end{bmatrix}\)

= \(\begin{bmatrix}5\\ 1\end{bmatrix}\)

 

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