Further Matrix Algebra

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Presentation transcript:

Further Matrix Algebra 18 January 2019 Def. e.g. Def. e.g. 2 x 2 In general e.g. 3 x 3 In general

Transpose of a matrix 18 January 2019 Ex

Transpose of a matrix 18 January 2019 Ex

Transpose of a matrix 18 January 2019 Ex

Further Matrix Algebra 18 January 2019 Res. Page 142 Exercise 6A

Further Matrix Algebra 18 January 2019 FP1 Def. Ex

Further Matrix Algebra 18 January 2019 FP1 Ex

Transpose of a matrix 18 January 2019 Ex

Transpose of a matrix 18 January 2019 Ex Page 146 Exercise 6B

Inverse Matrices 18 January 2019 FP1 FP1 FP1

Ex Inverse Matrices Minor Def. 18 January 2019 Def. Minor The minor of an element is the determinant of the elements which remain when the row and column containing the element are crossed out. Ex

Inverse Matrices Matrix of minors Def. 18 January 2019 Def. Matrix of minors The matrix of minors M of a matrix A is found by replacing each element of A with the minor of that element. Def. Matrix of cofactors

Inverse Matrices 18 January 2019 Def. Ex

Inverse Matrices 18 January 2019 Ex

Inverse Matrices 18 January 2019 Res. Proof

Inverse Matrices 18 January 2019 Ex Page 151 Exercise 6C

The domain or range of a function can have more than one dimension! Vector Functions 18 January 2019 Idea The domain or range of a function can have more than one dimension! Domain Range Example 1 1 1 2 3 1

The domain or range of a function can have more than one dimension! Vector Functions 18 January 2019 Idea The domain or range of a function can have more than one dimension! Domain Range Example 2 2 3 3

Linear Functions 18 January 2019 Linear Function Def. L1 L2 ? L1 L2

Linear Transformations 18 January 2019 Idea Ex

Linear Transformations 18 January 2019 Ex

Linear Transformations 18 January 2019 Ex

Linear Transformations 18 January 2019 Ex

Linear Transformations 18 January 2019 Ex Page 159 Exercise 6D

Linear Transformations 18 January 2019 Ex

Linear Transformations 18 January 2019 Ex

Linear Transformations 18 January 2019 Idea Don’t find the inverse matrix unless you have to. Ex Page 164 Exercise 6E

Eigenvalues and Eigenvectors 18 January 2019 Idea There will be some vectors for which the effect of a linear transformation is just like being multiplied by a scalar!

Eigenvalues and Eigenvectors 18 January 2019 Eigenvectors and Eigenvalues Def. Idea Finding eigenvalues

Eigenvalues and Eigenvectors 18 January 2019 Characterstic Equation Def. Idea Def. Normalised vector

Eigenvalues and Eigenvectors 18 January 2019 Ex

Eigenvalues and Eigenvectors 18 January 2019 Ex

Eigenvalues and Eigenvectors 18 January 2019 Ex

Eigenvalues and Eigenvectors 18 January 2019 Ex

Eigenvalues and Eigenvectors 18 January 2019 Ex Page 164 Exercise 6F

Diagonal Form 18 January 2019 Def. Orthogonal Res.

Diagonal Form Def. Res. Diagonal Matrix Diagonalisation 18 January 2019 Def. Diagonal Matrix Res. Diagonalisation

Diagonal Form 18 January 2019 Res. ?

Diagonal Form 18 January 2019 Ex

Diagonal Form 18 January 2019 Ex

Diagonal Form 18 January 2019 Ex

Diagonal Form 18 January 2019 Ex Page 186 Exercise 6G

? M1 Labels Ex Ex Def. Idea Reference to previous module 1 18 January 2019 M1 Reference to previous module 1 ? Quick Question Def. Definition Idea Key Idea Ex Example Ex Exercise