Linear Theorems

reduced row echelon form

The pivot in each nonzero row is equal to 1.

Each pivot is the only nonzero entry in its column.

Span

Ax=b has solution if and only if b is in the span of the columns of A

Matrix equivalency 1 ax=b

Ax=b has a solution in Rm, the span of the columns is Rm, a has a pivot in each row

Matrix Equivalency 2 ax=0

Ax=0 has a nontrivial solution, there’s a free variable, there’s a column without a pivot.

Span definition

the solution set of ax=0

Invertibility

(1) A −1 is invertible and its inverse is (A −1 ) −1 = A. (2) AB is invertible and its inverse is (AB) −1 = B −1A −1 . (3) A T is invertible and (A T ) −1 = (A −1 ) T . A transformation T is invertible if and only if it is both one-to-one and ont

subset

(1) The zero vector is in V. (2) If u and v are in V, then u + v is also in V. (3) If u is in V and c is in R, then cu is in V.

subspace/span

subspace is a span, and vice versa

triangular determinant

diagonal rows

invertible e value

can be invertible if 0 is not evalue

diagonizability

only diagonalizable if similar to diagonal matrix

linear independent eigenvectors

projection

linear transformation