math 18

onto

pivot in every row (consistency)

one-to-one

pivot in every col (no free variables)

Col A =

{ pivot columns of A } (of the og matrix, not ref A)

Row A =

{ non zero rows of ref A }

Nul A =

{ vectors of the free variable coefficients } OR solution set of Ax = 0

Kernel of transformation T =

Nul A (all inputs that output 0)

Range of transformation T =

Col A (all possible outputs)

rank A =

num of pivot positions in A

Nullity A =

number of free variables in Nul A

Rank-Nullity Theorem

rank A + nullity A = # of columns in A

finding eigenvalues

solve equation det(A - λI) = 0 for λ

finding eigenvectors

solve equation (A - λI)x = 0

finding eigenspace

Nul(A - λI) / (A - λI)x = 0 (solution set of x)

diagonalization

A = PDP^-1

P matrix =

matrix of eigenvectors of A

D matrix =

diagonal matrix of eigenvalues

a matrix is NOT diagonizable if:

the eigenvectors do not span the dimension of the matrix

distance with dot product

|| v - u ||

for a basis to be orthogonal, it must be:

1) every vector is orthogonal to one another

2) be a spanning set

3) be linearly independent

for a basis to be orthonormal, it must be:

1) every vector is orthogonal to one another

2) every vector has a norm of 1 (||v|| = 1)

2) be a spanning set

3) be linearly independent