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Chapter 9: Change of Basis

Chapter 9: Change of Basis

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Chapter 9: Change of Basis
9.1 The Matrix of a Linear Transformation
Definition 9.1 Coordinate Vector C_B(\mathbf{v}) of v for a basis B
Example 9.1.1
Theorem 9.1.1
Definition 9.2 Matrix M_{DB}(T) of T: V \to W for bases D and B
Theorem 9.1.2
Example 9.1.2
Example 9.1.3
Example 9.1.4
Example 9.1.5
Theorem 9.1.3
Theorem 9.1.4
Theorem 9.1.5
Example 9.1.6
Example 9.1.7
Exercises for 9.1
9.2 Operators and Similarity
The B-matrix of an Operator
Definition 9.3 Matrix MDB(T ) of T : V W for basis B
Theorem 9.2.1
Definition 9.4 Change Matrix 1 0 \% = 1 0 for bases B and D
Theorem 9.2.2
Example 9.2.1
Theorem 9.2.3: Similarity Theorem
Example 9.2.2
Solution. We have
Theorem 9.2.4
Proof.
Example 9.2.3
Example 9.2.4
Solution.
Example 9.2.5
Example 9.2.6
Example 9.2.7
Exercises for 9.2
9.3 Invariant Subspaces and Direct Sums
Definition 9.5 T -invariant Subspace
Example 9.3.1
Example 9.3.2
Example 9.3.3
Example 9.3.4
Definition 9.6 Restriction of an Operator
Theorem 9.3.1
Theorem 9.3.2
Example 9.3.5
Eigenvalues
Theorem 9.3.3
Example 9.3.6
Theorem 9.3.4
Direct Sums
Definition 9.7 Direct Sum of Subspaces
Example 9.3.7
Example 9.3.8
Example 9.3.9
Theorem 9.3.5
Theorem 9.3.6
Theorem 9.3.7
Definition 9.8 Reducible Linear Operator
Example 9.3.10
Solution.
Example 9.3.11
Exercises for 9.3
Exercise 9.3.5
Exercise 9.3.33
Section worksheets