## Matrix multiplication and linear transformations

Let $$A$$ be a $$p\times n$$ matrix and $$B$$ be a $$m\times p$$ matrix. Define the product $$BA$$ to be the matrix given by $$\begin{bmatrix} BA_1 & BA_2 & \cdots & BA_n\end{bmatrix}$$ where $$A_i$$ denotes the $$i$$th column of $$A$$.

If $$S$$ and $$T$$ are linear transformations given by $$S(x) = Ax$$ and $$T(x)= Bx$$, then $$T(S(x)) = (BA)x$$.

Matrix multiplication is defined in such a convoluted manner precisely to represent composition of linear transformations.

### Examples

1. Let $$A = \begin{bmatrix} 1 & 2\\3 & 5\end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 0\\ -1 & 1\end{bmatrix}$$.

Then $$BA_1 = \begin{bmatrix} 1 & 0\\ -1 & 1\end{bmatrix} \begin{bmatrix} 1 \\3 \end{bmatrix}= \begin{bmatrix} 1(1) + 0(3)\\ -1(1) + 1(3) \end{bmatrix}= \begin{bmatrix} 1\\ 2 \end{bmatrix}$$, and $$BA_2 = \begin{bmatrix} 1 & 0\\ -1 & 1\end{bmatrix} \begin{bmatrix} 2 \\5 \end{bmatrix}= \begin{bmatrix} 1(2) + 0(5)\\ -1(2) + 1(5) \end{bmatrix}= \begin{bmatrix} 2\\ 3 \end{bmatrix}$$. Hence, $$BA = \begin{bmatrix} 1&2\\ 2 &3 \end{bmatrix}$$.

2. Let $$A = \begin{bmatrix} 1 & 2 & 3\end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 4\\ -1 & 1 \\ 2 & -2\end{bmatrix}$$.

Then $$AB_1 = \begin{bmatrix} 1 & 2 & 3\end{bmatrix} \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1(1) + 2(-1) + 3(2) \end{bmatrix} = \begin{bmatrix} 5\end{bmatrix}$$, and $$AB_2 = \begin{bmatrix} 1 & 2 & 3\end{bmatrix} \begin{bmatrix} 4 \\ 1 \\ -2 \end{bmatrix} = \begin{bmatrix} 1(4) + 2(1) + 3(-2) \end{bmatrix}= \begin{bmatrix} 0\end{bmatrix}$$. Hence, $$AB = \begin{bmatrix} 5 & 0 \end{bmatrix}$$.

3. Let $$A = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$$ and $$B = \begin{bmatrix} 3 & 5\end{bmatrix}$$.

Then $$AB_1 = \begin{bmatrix} 1 \\ -2 \end{bmatrix} \begin{bmatrix} 3 \end{bmatrix} = \begin{bmatrix} 3 \\ -6 \end{bmatrix}$$, and $$AB_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix} \begin{bmatrix} 5 \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$$. Hence, $$AB = \begin{bmatrix} 3 & 5 \\ -6 & -10 \end{bmatrix}$$.

## Beware!

Note that the matrix product $$AB$$ is defined only when the number of columns of $$A$$ matches the number of rows of $$B$$.

Also, in general $$AB$$ and $$BA$$ are not necessarily equal. For instance, it is possible that the product $$AB$$ is defined while $$BA$$ is not (when the number of columns of $$B$$ is not the same as the number of rows of $$A$$.)

### Examples

1. Let $$A = \begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix}$$. Let $$B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$. Then $$AB = \begin{bmatrix} 0 & 1 \\ 0 & 2 \end{bmatrix}$$ but $$BA = \begin{bmatrix} 2 & 0 \\ 1 & 0\end{bmatrix}$$.

2. Let $$A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$$. Let $$B = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$. Then $$AB = \begin{bmatrix} 1 \\ 2\end{bmatrix}$$ but the product $$BA$$ cannot be formed since the number of columns of $$B$$ is different from the number of rows of $$A$$.

## Exercises

1. Compute each of the following matrix products.

1. $$\begin{bmatrix} 1 & 3 \\ 2 & 4\end{bmatrix} \begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}$$

2. $$\begin{bmatrix} 0 & 1 & 2 & -1 \\ 4 & 3 & 0 & 2\end{bmatrix} \begin{bmatrix} -1 \\ 0 \\ 1 \\ 3\end{bmatrix}$$

3. $$\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} -1 & 1 \end{bmatrix}$$

2. Let $$A = \begin{bmatrix} 1 & 2 \end{bmatrix}$$, $$B = \begin{bmatrix} 1 & 4\\ -1 & 1 \end{bmatrix}$$, $$C = \begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}$$.

1. Compute $$AB$$ and $$BC$$.

2. Using the results of the previous part, compute $$(AB)C$$ and $$A(BC)$$. What can you say about these products?