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矩阵乘法

1. Introduction

Introduction

2. Composition of linear transformations

Composition of linear transformations

3. Matrix multiplication

Matrix multiplication

4. Composition and Matrix Multiplication

Composition and Matrix Multiplication

5. The algebra of transformations and matrices

The algebra of transformations and matrices

6. Size of a matrix product

Let $A$ be a $2\times 4$ matrix and let $B$ be a $4\times 7$ matrix. What is the size of the product $AB$ ?

7. Order matters

Let

A=\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right)

and

B=\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right).

Which statement is true?

8. Composition and matrix product

Let $T$ and $U$ be linear transformations with standard matrices $A$ and $B$ . The standard matrix of the composition $T\circ U$ is the ______ of $A$ and $B$ . Enter one word.

9. One entry of a product

Let

A=\left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right)

and

B=\left(\begin{array}{cc} 1 & -3 \\ 2 & -2 \\ 3 & -1 \end{array}\right).

Enter the entry in the first row and second column of $AB$ (a number).

10. Product of symmetric matrices

Let

A=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right),\qquad B=\left(\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{array}\right),

which are both symmetric. Enter the entry in the first row and second column of the product $AB$ (a number).

11. Inner dimensions must match

The matrix product $AB$ is defined, and $B$ is a $4\times 7$ matrix. What can you conclude about $A$ ?

12. Compose through a shear

Let

A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

and $x = (2,3)$ . Compute $A(Ax)$ and enter the result as $x,y$ .

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