Up Main

Example 1

Let \(T\) be a linear transformation given by \(T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right) = \begin{bmatrix} u + v - w \\ 2u + w\end{bmatrix}\). Find a matrix \(A\) such that \(T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right) = A \begin{bmatrix} u \\ v \\ w\end{bmatrix}\).

Example 2

Let \(T\) be a linear transformation given by \(T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right) = \begin{bmatrix} u - v + w \\ 3u + v - w\end{bmatrix}\). Let \(S\) be a linear transformation given by \(S\left(\begin{bmatrix} x\\ y\end{bmatrix}\right) = \begin{bmatrix} 2x + y \\ -x + 2y\end{bmatrix}\). Find a matrix \(A\) such that \(S\left(T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right)\right) = A \begin{bmatrix} u \\ v \\ w\end{bmatrix}\).

Example 3

Compute the matrix product \(\begin{bmatrix} 1 & 2 \\ -1 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix} 3 & 1 & 2 \\ -1 & 0 & 4\end{bmatrix}\).

Example 4

Let \(A = \left[\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 1 & 3 \\ -2 & 1 & 0\\ \end{array}\right]\) and let \(B = \left[\begin{array}{rrr} 1 & -1 & 1\\ 0 & 3 & 9 \\ 0 & -1 & 2\\ \end{array}\right]\). Find a matrix \(M\) such that such that \(MA = B\).

Example 5

Let \(A = \begin{bmatrix} 1 & 0 & 2\\ 1 & 3 & 3\\ 0 & 1 & 0\end{bmatrix}\). Find all solutions to the systems \(Ax = e_i\) for \(i = 1,2,3\) where \(e_i\) denotes the \(i\)th column of the \(3\times 3\) identity matrix.