math 22

Theorum 1 (1.5) Ax = 0 has a non trivial solution if and only if…

the corresponding system has at least 1 free variable

Theorum 4 (1.5)

Let A be an m * n matrix.

The following statements are true

a. Ax = b has a solution for each b∈Rm

b. each b∈Rm is a linear combo of the columns

c. Span {cols of A} = Rm

d. A has a pivot in each row (justifies the rest of these)

homogenous system

the solution for each row is 0

A set S = {v1, v2 … vp} for p >= 2 is linearly dependent if and only if…

at least one of the vectors in S is a linear combination of the others

Theorem 7 (1.7): Any set {v1, v2 … vp} in Rn is linearly dependent if

p > n

there are more vectors than length of the vector

2.1 theorem 2 (Properties of Matrix Multiplication)

2.1 theorem 1

2.1 theorem 3 (transpose of a matrix)

transpose of a matrix

taking the original matrix and flipping it across its main diagonal (the line running from the top-left to the bottom-right)

2.2 th 4: how to find inverse of 2 × 2 matrix

2.2 th. 5: If A is an invertible n x n matrix, then ______

for each b in Rn , the equation Ax = b has the unique solution x = A^-1 * b.

2.2 Th 6 a: If A is an invertible matrix, _____

A^-1 is invertible and (A^-1)^-1 = A

2.2 Th 6 b: If A is an invertible matrix, then ____

1) so is AB, and

2) the inverse of AB is the product of the inverses of A and B in the reverse order.

That is, (AB)^-1 = B^-1*A^-1

2.2 Th 6 c: If A is an invertible matrix, then so is A^T, and _____

the inverse of A ^T is the transpose of A ^-1 .

That is, (A^T)^-1 = (A^1)^ T

2.2 Th 7: An n x n matrix A is invertible if and only if ….

A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A^-1 .

pivot in every row means…

onto/spans

pivot in every column means….

linearly independent (no free vars)

2.1 Th 1: obvious operations about matrices

2.1 Th 2: Properties of matrix multiplication

2.1 Th 3: properties of transposition

how does transposition work?

r1 becomes c1, r2 becomes c2, r2 becomes c3 etc

4.1 Th 1: A span of any 2 vectors in Rn….

is a subspace of Rn