Math Review Guide.pdf - Math 54 review The following is a...

Math 54 review
The following is a (possibly incomplete rough draft of a) list of key
points from the class. Making sure you know all of these topics and can
solve problems using them should be good preparation for the final exam.
Linear algebra chapter 1.
• How to solve a system of linear equations by row reduction. Pivot
variables and free variables.
• How to multiply a matrix by a vector, either by taking dot products
with the rows, or by taking a linear combination of the columns.
• The definition of linear independence.
• The definition of a linear transformation.
Linear algebra chapter 2.
• How to add and multiply matrices.
• Associative and distributive properties of matrix multiplication. Ma-
trix multiplication is (usually) not commutative!
• Equivalence between m × n matrices and linear transformations from
R
n
to R
m
. Matrix multiplication corresponds to composition of linear
transformations.
• If A is an m × n matrix, then T
A
is injective if and only if the columns
of A are linearly independent, or equivalently every column contains
a pivot, which requires m ≥ n ; and T
A
is surjective if and only if the
columns of A span R
m
, or equivalently every row contains a pivot,
which requires m ≤ n .
• A and B are inverses of each other if both AB and BA equal the
identity.
• How to test whether a matrix is invertible and how to compute the
inverse when it exists, either by row reduction (in the general case) or
Cramer’s rule (in the 2 × 2 case).
1

• Matrices which are not square cannot be invertible. The reason is that
if A is an m × n matrix and AB = I then T
A
is surjective and T
B
is
injective, both of which require m ≤ n . If BA = I also, then likewise
n ≤ m .
• If A and B are n × n matrices with AB = I , then BA = I also.
Linear algebra chapter 3.
• How to calculate the determinant of a square matrix, either by row
reduction, by expanding along a row or column, or as a sum over
permutations (at least for 2 × 2 and 3 × 3).
• When A is a square matrix, A is invertible if and only if det( A ) 6 = 0.
• The geometric meaning of determinants: if A is an n × n matrix and U
is a region in R
n
with finite volume, then vol( T
A
( U )) = | det( A ) | vol( U );
and det( A ) > 0 when A preserves orientation.
Linear algebra chapter 4.
• Definition of a vector space. Main examples: R
n
, subsets of a vector
space, and some function spaces.
• Definition of a subspace of a vector space. Examples: span of a subset;