linear test 4

How to find preimage

Put into a linear system and solve then back sub to get the other variable

How to do Eigenvalues

A-λI and make determinate =0

Eigenvectors

Plug values in and to A-λI and solve

Eigenvalues

A-λI

Diagonalization

P^-1(AP)

How to find image

plug points into linear transformation definition

Isomorphic

Dimension of matrix is same a R^#

T(u+v)

T(u)+T(v)

T(cu)

cT(u)

T(v)=Av

Can be true

MxN

R^m and R^n

Kernel

Vectors that satisfy T(v)=0

Ax=0 solution space

Range

T(x)=Ax

And is column space of Ax

Nullity

dimension of Kernel

Rank

Dimension of range

Dimension (domain)

dim(range)+dim(kernel)

One to one

ker(T)=0

Onto

Rank(T)=dim(W)

Rank(T)=n

One to one

Rank(T)=m

Onto

W&V same dimension of N

One to one and onto

Isomorphic if

V&W are same direction