The general form of a linear transformation is given by the following assignment where a, b, c, d, e, f are real numbers:

To be able to interpret the geometric meaning of a, b, c, d, we shall set e = f = 0, i.e., for the moment we exclude translations in the transformation. In this case, we have

Again, there is a more convenient notation for expressing these two equations as one:

The array of the four numbers a, b, c, d arranged in a square is called a matrix. It has dimension 2 x 2 (read: two by two) since there are two horizontal rows of numbers and two vertical columns of numbers. The operation being performed is called multiplication of the matrix by the coordinate pair. It is convenient to encode the transformation in this single object, the matrix.

Try computing the following transformations by multiplying the given 2 x 2 transformation matrix by the given coordinate pair.

Practice
Computing Transformations

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In the window above multiply the vector by the matrix and type your result in the empty textfields. Click "Submit" to submit your answer. If your answer is incorrect, after three tries the result will be given to you. Click "Next" to try another matrix and vector combination.