Vectors and matrices

Matrices as linear functions - writing a linear function in vector form as

Linear combinations of vectors and the matrix-vector product

where the matrix

A has the vectors as its columns.
Conversely: is a combination of the columns

of A with coefficients from

Matrix operations - addition, scalar multiplication, multiplication, and their
properties.

Basic matrix algebra. The individual elements of AB as row-column products:

Every column of AB is a combination of the columns of A; every row of AB is a
combination

of the rows of B.

Generating AB column by column: ;

Generating AB row by row: row i of

Matrix transpose,

Solution of linear systems:

The matrix form of a linear system.

Gauissian elimination to row echelon form, and then to row reduced echelon form.

What row echelon form of A tells you:

the rank, whether solutions always exist, whether solutions are unique

What row echelon form of tells you: whether
a solution of exists, also tells

you the row echelon form of A.

Pivot variables and free variables.

How to write the general solution of of
in vector form from the row reduced

echelon form of :

the long way: introduce a parameter for each free variable, solve for each pivot
variable

via the equations in the row reduced echelon form, write your answer in vector
form

the short way: generate the basic solutions of the homogeneous system
using

the row reduced echelon form of A, by in turn setting each free variable to 1
and the others

to zero; then generate a particular solution of
using the row reduced echelon form of

by setting all the free variables to zero.
Add the general solution of
,

obtained via a general linear combination of the basic solutions of
, to the particular

solution of
to get the general solution

Generating a basic set of solutions of

Linear independence: The definition of linear
independence. The connection to

homogeneous systems of linear equations. Determining linear independence of a
set of

vectors from their row echelon form - determining specific linear combinations
from the row

reduced echelon form: expressing one vector as a linear combination of other
vectors,

finding a nontrivial linear combination of vectors that gives the zero vector.

Rank of a matrix: Connection of the rank of A with linear systems
, specifically how

the rank of A tells us whether solutions always exist, whether solutions are
unique;

connection of rank of A with linear independence of columns of A.

Determinants:

Defining properties of the determinant function

Properties of the determinant:

Effect of row operations on value of the determinant

What the determinant determines: detA ≠ 0 if and only if A ~ I

Determinant of a triangular matrix

Using row operations to find detA

"Pulling out" common factors in a row or column

Interchanging rows

detA^{T} = detA

The Laplace expansion of detA, the i, j cofactor

Evaluating a determinant using "pivotal condensation": Successively using
row/column

elimination steps so that there is only one nonzero entry in a given column/row
and then

expanding.

Determinants with variable entries

Equivalences:

detA ≠ 0

A ~ I (the row reduced echelon form of A is I )

has a solution for
each

has unique solutions

has only
as a solution

rank A = n

A is invertible - A^{-1} exists - A is nonsingular (these all mean the same thing)

the columns of A are linearly independent

The inverse of a matrix: (either one implies the other for
square

matrices)

Calculating A ^{-1} using row operations:
. Note that if A ~ I is not
true, then

A ^{-1} does not exist.

The formula for A^{-1} in terms of cofactors:
where C is the cofactor matrix.

Cramer’s rule:

If
with detA ≠ 0 then
where the numerator is the

determinant of the matrix obtained by replacing column j of A by
, so that
represents

column 1 of A and so forth.

Checking an inverse calculation.

Given A^{-1} , the unique solution of
is
(multiply both sides on the left by A^{-1}

to see that must be true)