Change Of Basis Flashcards
What is the change of basis matrix from alpha to alpha’
Change of basis matrix from alpha to alpha’ is:
V’j = sum(Pij*Vj) where P is an n x n matrix
P is transpose of matrix used to get from alpha to alpha’
When are 2 matrices A, A’ equivalent
2 matrices A, A’ are equivalent when:
E Q and P inheritable matrices such that A’ = Q^-1AP
Matrices representing same linear map are equivalent
When are 2 matrices A, A’ similar
2 matrices A, A’ are similar if:
E invertiblr matrix P such that: A’ = P^-1AP
Matrices representing same linear operator are similar
What is a pivot
Pivot is the 1st non-zero entry of each row of an REF matrix, columns in which they appear are pivot columns
What is the rank of linear map phi equal to
Rank of linear map phi is equal to dimension of image of phi (number of pivot columns in matrix A representing phi)
What is inverse of matrices in REF form with lambda in middle
Inverse of matrices in REF form with lambda in middle is same but 1/lambda in middle
What is inverse of matrices in REF with lambda top middle and 1s in leading diagonal
inverse of matrices in REF with lambda top middle and 1s in diagonal is same but -lambda in top middle
If V is isomorphic to W, what does this mean for the dimensions
If V is isomorphic to W, this means dim V = Sim W
If 2 matrices A and B are equivalent, what are dimensions and rank
If 2 matrices A and B are equivalent, then dim A = dimB
Rank A = rank B
What is change of basis matrix (Jennifer)
Change of basis matrix I (Jennifer) is:
Matrix with other basis vectors as columns multiplied by vectors in other coordinate system gives same vector in our coordinate system
How to find determinants of upper and lower triangular matrices
Find determinants of upper and lower triangular matrices by multiplying along diagonals
Upper triangular, entries below diagonal must = 0
Lower triangular, entries above diagonal must = 0
