This document is a list of some material in linear algebra
that you should be familiar with.
Throughout, we will take A to be the 3 x 4 matrix
I assume you are familiar with matrix and vector addition and multiplication.
 All vectors will be column vectors.
 Given a vector v,
if we say that ,
we mean that v has at least one nonzero
component.
 The transpose of a vector or matrix is denoted by a superscript T.
For example,
 The inner product or dot product of two vectors u and v
in
can be written u^{T}v; this denotes
.
If u^{T}v=0 then u and v are orthogonal.
 The null space of A is the set of all solutions x to the
matrixvector equation Ax=0.
 To solve a system of equations Ax=b, use Gaussian elimination.
For example, if
,
then we solve Ax=b as follows:
(We set up the augmented matrix and row reduce (or pivot) to upper
triangular form.)
Thus, the solutions are all vectors x of the form
for any numbers s and t.
 The span of a set of vectors is the set of all linear combinations
of the vectors. For example, if
and
then the span of v^{1} and v^{2} is the set of
all vectors of the form sv^{1}+tv^{2} for some scalars s and t.
 The span of a set of vectors in
gives a subspace of .
Any nontrivial subspace can be written as the span of any one of
uncountably many sets of vectors.
 A set of vectors
is linearly independent
if the only solution to the vector equation
is
for all i.
If a set of vectors is not linearly independent,
then it is linearly dependent.
For example, the rows of A are not linearly independent,
since
To determine whether a set of vectors is linearly independent,
write the vectors as columns of a matrix C, say, and solve Cx=0.
If there are any nontrivial solutions then the vectors are linearly
dependent; otherwise, they are linearly independent.
 If a linearly independent set of vectors spans a subspace
then the vectors form a basis for that subspace.
For example, v^{1} and v^{2} form a basis for the span of the rows of A.
Given a subspace S, every basis of S contains the same number of
vectors; this number is the dimension of the subspace.
To find a basis for the span of a set of vectors,
write the vectors as rows of a matrix and then row reduce the matrix.
 The span of the rows of a matrix is called the row space of the
matrix.
The dimension of the row space is the rank of the matrix.
 The span of the columns of a matrix is called the range or the
column space of the matrix.
The row space and the column space always have the same dimension.
 If M is an m x n matrix then the null space and the row space of M
are subspaces of
and the range of M is a subspace of .
 If u is in the row space of a matrix M and v is in the null space
of M then the vectors are orthogonal.
The dimension of the null space of a matrix is the nullity
of the matrix.
If M has n columns then rank(M)+nullity(M)=n.
Any basis for the row space together with any basis for the null space
gives a basis for .
 If M is a square matrix,
is a scalar, and x is a vector
satisfying
then x is an eigenvector
of M with corresponding eigenvalue .
For example, the vector
is an eigenvector of the matrix
with eigenvalue .
 The eigenvalues of a symmetric matrix are always real.
A nonsymmetric matrix may have complex eigenvalues.
 Given a symmetric matrix M,
the following are equivalent:
 1.
 All the eigenvalues of M are positive.
 2.
 x^{T}Mx>0 for any .
 3.
 M is positive definite.
 Given a symmetric matrix M,
the following are equivalent:
 1.
 All the eigenvalues of M are nonnegative.
 2.

for any x.
 3.
 M is positive semidefinite.
John E. Mitchell
20040831