Matrix Multiplication

Author:Ku, Yin Bon (Albert)

Composition of Linear Transformations

Suppose

and

are two linear transformations. We consider their composition

i.e. for any vector

. You can verify that

is again a linear transformation. A natural question: How can we write down the matrix for

if we are given the matrices for

and

? The matrix for

is formed by the column vectors

and

. Suppose

and

be the matrices for

and

respectively, we have

Hence the matrix for

is defined to be the matrix multiplication of and as follows:

In the following applet, matrix multiplication of 2 x 2 matrices is visualized as a composition of linear transformations. Try the following tasks: 1. Find

and

such that

. 2. Find

and

such that

, and

. 3. Find

and

such that

but