Domain Range definition: T is a linear transformation, EIGENVECTOR EIGENVALUE.

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Presentation transcript:

Domain Range

definition: T is a linear transformation, EIGENVECTOR EIGENVALUE

A is the matrix for a linear transformation T relative to the STANDARD BASIS

T T

T T The matrix for T relative to the  basis

T T Eigenvectors for T Diagonal matrix

The matrix for a linear transformation T relative to a basis of eigenvectors will be diagonal

To find eigenvalues and eigenvectors for a given matrix A: Solve for and A = A = I A = I - A ) = I - (

To find eigenvalues and eigenvectors for a given matrix A: Solve for and A ) = I - ( Remember: is a NONZERO vector in the null space of the matrix: A )I - (

is a NONZERO vector in the null space of the matrix: A )I - ( The matrix has a nonzero vector in its null space iff: A )I - ( I - ( det = 0

A )I - ( det = 0 A =I - -

A )I - ( det =

A )I - ( det =

A )I - ( det = This is called the characteristic polynomial

A )I - ( det = = 0 the eigenvalues are 2 and -1

A )I - ( = the null space of 2I - A = 2 the eigenvectors belonging to 2 are nonzero vectors in the null space of 2I - A

A )I - ( = the null space of -1I - A = the eigenvectors belonging to -1 are nonzero vectors in the null space of -1I - A

Matrix for T relative to standard basis

Matrix for T relative to columns of P

Basis of eigenvectors

eigenvalues