Question

Find a matrix representation of the transformation L(x, y) = (3x + 4y, x − 2y).

Answer 1

`\left[\begin{array}{cc}x&y\end{array}\right] * \left[\begin{array}{cc}3&1\\4&-2\end{array}\right] = \left[\begin{array}{cc}3x+4y&x-2y\end{array}\right]`

Explanation:

The general matrix representation for this transformation would be:

`\left[\begin{array}{cc}x&y\end{array}\right] * A = \left[\begin{array}{cc}3x+4y&x-2y\end{array}\right]`

As the matrix A should have the same amount of rows as columns in the firs matrix and the same amount of columns as the result matrix it should be a 2x2 matrix.

`\left[\begin{array}{cc}x&y\end{array}\right] * \left[\begin{array}{cc}a&b\\c&d\end{array}\right] = \left[\begin{array}{cc}3x+4y&x-2y\end{array}\right]`

Solving the matrix product you have that the members of the result matrix are:

3x+4y = a*x + c*y

x - 2y = b*x + d*y

So the matrix A should be:

`\left[\begin{array}{cc}3&1\\4&-2\end{array}\right]`