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Matrix Transformations

1. Linear Transformations and Matrix Algebra

Linear Transformations and Matrix Algebra

2. Introduction

Introduction

3. Matrices as Functions

Matrices as Functions

4. Transformations

Transformations

5. Matrix Transformations

Matrix Transformations

6. Domain and codomain

Let $A$ be a $3\times 2$ matrix and let $T(x)=Ax$ be the associated matrix transformation. Which statement is correct?

7. Range of a matrix transformation

Let

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

and let $T(x)=Ax$ . What is the range of $T$ ?

8. Name the range

Let $A$ be an $m\times n$ matrix and let $T(x)=Ax$ . The range of $T$ is the ______ of $A$ . Enter two words.

9. Evaluate a matrix transformation

Let

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

and let $T(x)=Ax$ . Enter $T(u)$ for $u=(-1,-2,-3)$ as two numbers separated by commas (for example: 1,-2).

10. Spring extensions in a chain

A chain of four masses joined by five springs is described by the $5\times 4$ matrix

A=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{array}\right),

whose transformation $T(u)=Au$ turns the displacements $u$ of the four masses into the extensions of the five springs. For $u=(2,5,1,3)$ enter $T(u)$ as five numbers separated by commas (for example: 1,2,3,4,5).

11. In your own words

In your own words, state the single most important definition or theorem of this section, and explain what it says.

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