Chapter 7: Linear Transformations

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7.1 Examples and Elementary Properties

Definition 7.1 Linear Transformations of Vector Spaces

Example 7.1.1

Example 7.1.2

Example 7.1.3

Example 7.1.4

Theorem 7.1.1

Proof.

Example 7.1.5

Theorem 7.1.2

Example 7.1.6

Theorem 7.1.3

Example 7.1.7

Exercises for 7.1

7.2 Kernel and Image of a Linear Transformation

Definition 7.2 Kernel and Image of a Linear Transformation

Example 7.2.1

Theorem 7.2.1

Example 7.2.2

Example 7.2.3

One-to-One and Onto Transformations

Definition 7.3 One-to-one and Onto Linear Transformations

Theorem 7.2.2

Example 7.2.4

Example 7.2.5

Example 7.2.6

Theorem 7.2.3

Proof.

The Dimension Theorem

Theorem 7.2.4: Dimension Theorem

Theorem 7.2.5

Example 7.2.7

Example 7.2.8

Example 7.2.9

Example 7.2.10

Example 7.2.11

Exercises for 7.2

7.3 Isomorphisms and Composition

Definition 7.4 Isomorphic Vector Spaces

Example 7.3.1

Example 7.3.2

Example 7.3.3

Theorem 7.3.1

Theorem 7.3.2

Corollary 7.3.1

Corollary 7.3.2

Example 7.3.4

Theorem 7.3.3

Composition

Definition 7.5 Composition of Linear Transformations

Example 7.3.5

Theorem 7.3.4: 3

Theorem 7.3.5

Example 7.3.6

Example 7.3.7

Example 7.3.8

Example 7.3.9

Exercises for 7.3

7.4 A Theorem about Differential Equations

Theorem 7.4.1

Lemma 7.4.1

Lemma 7.4.2

Lemma 7.4.3: Kernel Lemma

7.5 More on Linear Recurrences

Example 7.5.1

Lemma 7.5.1

Theorem 7.5.1

Example 7.5.2

Theorem 7.5.2

Example 7.5.3

The Shift Operator

Theorem 7.5.3

Theorem 7.5.4

Example 7.5.4

Remark 1

Remark 2

Exercises for 7.5

Section worksheets