Math 311101 — Test I Review — Summer
I, 2016
General Information
Test 1 (Wednesday, June 15) will have 6 to 8 questions, some with
multiple parts. It will cover sections 1.11.5, 2.12.2, 3.13.4, and
3.6, in the text. In addition, it will include material from these
sets of
notes:
Notes on Row Reduction; and, Methods
for Finding Bases. Problems will be similar to ones done for
homework or examples done in class or in the sets of notes.
Practice tests may be found at theese links:
Fall 2003,
Fall 2004 and
Summer 2006. I will have office hours on
Monday, 11:4512:45, Tuesday afternoon, 11:45 am2:30 pm, and
Wednesday morning, 9:159:45 am.

Calculators. You may use scientific calculators to do numerical
calculations — logs, exponentials, and so on. You
may not use any calculator that has the capability of doing
algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
Topics Covered
Systems & matrices
 Linear systems and reduced row echelon form of a matrix
 Know the rowreduction algorithm as outlined in my
Notes on Row Reduction. Be able to put a matrix in reduced row
echelon form. Be able to identify the lead variables
and free variables. Be able to solve linear systems via row
reduction.
 Special types of systems: homogeneous, overdetermined,
underdetermined.
 Homogeneous systems. Know the connection with solutions to a
general system and the corresponding homogeneous system.
 Be able to solve for currents and voltages in electrical
networks.
 Matrices
 Matrix algebra. Sum, product, scalar multiples, row vectors,
column vectors, transpose, symmetric matrix, identity matrix, zero
matrix, size of a matrix, (i,j) entry, notation. Know the "basic matrix trick"
Ax = x_{1}a_{1} + x_{2}
a_{2} + ... +x_{n} a_{n}
where the a_{j}'s are the columns of A.
 Inverse of a matrix.
 Be able to be find the inverse of a matrix
or show that a matrix is singular via row reducing [AI].
 Know that these are equivalent conditions for A to be invertible:
 A is nonsingular. (That is, A^{−1} exists.)
 Ax = 0 has only x = 0 as a
solution.
 A is row equivalent to I.
 det(A) ≠ 0.
 The columns of A are linearly independent.
 Know that these are equivalent conditions for A to be singular:
 A is singular. (That is, A^{−1} doesn't exist.)
 Ax = 0 has a nontrivial solution x
≠ 0 as a solution.
 A is row equivalent to a matrix with 0's in the last row.
 det(A) = 0.
 The columns of A are linearly dependent
 Elementary matrices.
 Know the three types of elementary matrices and how they
correspond to row operations. Be able to solve problems similar to
those assigned. Be able to write A or A^{−1} in terms of
elementary matrices.
Determinants
 Basic properties. Know the basic properties for
determinants. Be able to calculate the determinant of a matrix via its
cofactor expansion about a row or a column, or by using row operations,
or by some combination of the two methods.
 Determinants of special matrices. The determinant of an
upper triangular, lower triangular, or diagonal matrix is the product
of the diagonal entries.
 Row reduction of a matrix A and det(A). Know how det(EA)
is related to det(A) for an elementary matrix. Know the determinant
of the three types of elementary matrices. Be able to read off the
determinant of a matrix from the row operations used to reduce it
and its row echelon form.
 Inverses. Be able to determine whether an n×n
matrix A is invertible from knowing det A.
 Product rule. det(AB)=det(A)det(B).
Vector spaces
 Properties and examples.
 Closure axioms. Addition: If u and v are
vectors, then so is u + v. Multiplication by
scalars: If c is a scalar and v is a vector, then
c·v is a vector.
 Special vector spaces: R^{n},
R^{m×n} (m×n matrices), P_{n}
(polynomials of degree less than n), C[a,b] (continuous functions
on [a,b]), C^{(n)}[a,b] (n times continuously differentiable
functions on [a,b]).
 Subspaces
 Know the test to determine whether a subset S of a vector space
is a subspace: (i) Is 0 in S? (ii) Is S closed under vector
addition? (iii) Is S closed under multiplication by a scalar? (Instead
of (i) and (iii), test whether, for arbitary scalars α and
β, and arbitrary vectors u and v in S,
αu + βv is in S.)
 Know what inear combinations, spans, and spanning sets are. Be
able to determine whether or not a set is a spanning set for a vector
space. Be able to show that the span of a set of vectors is a
subspace.
 Null space of a matrix A, N(A) = {x in
R^{n}  Ax = 0}.
 Row space of A. This is the span of the rows of A.
 Column space of A. This is the span of the columns of A.
 Linear Independence and Linear Dependence
 Definition and test for LI and LD sets of vectors. To test
whether a set S =
{v_{1}, v_{2}, ...,
v_{k}) is LI or LD, start with the homogeneous equation
(∗) c_{1}v_{1} +
c_{2}v_{2}, ...,
c_{k}v_{k} = 0
 IF the only scalars for which the equation (∗) hold are
c_{1} = c_{2} = ... = c_{k} = 0, then S is
LI.
 IF there are nonzero scalars for which (∗) holds are, then
S is LD.
 "Coordinate theorem." (Theorem 3.2 in Leon.) Be able to prove
this.
 Basis, Dimension and Coordinates
 Definition of basis. A set S = {v_{1},
v_{2}, ..., v_{n}} is a basis for a
vector space V if and only if (i) S is LI and (ii) S spans V.
.
 Definition of dimension. If V has a basis with n>0 vectors in
it, then dim(V) = n. If V ={0}, dim(V) = 0. If V has LI
arbitrarily large LI sets in it, dim(V) is infinity.
 Know the standard bases for P_{n},
R^{n}, R^{m×n}
 Suppose that a vector space V has dim(V) = n and S = {v_{1},
v_{2}, ..., v_{m}}.
 m < n
 S cannot span V.
 If S is linearly independent and m < n, then vectors may be
added to S to make it into a basis.
 m > n
 If m > n, then S is linearly dependent.
 If S m > n and spans V, then S may be pared
down to be a basis for V.
 m = n
 If m = n and S is linearly independent, than S spans V and is a basis.
 If m = n and S spans V, then S is linearly independent and is a
basis.
 Null, Row, and Column Spaces
 Null space. N(A) = {x in
R^{n}  Ax = 0}.
 Row space. This is the span of all of the rows of A,
Span(r_{1}, r_{2}, ...,
r_{m}).
 Column space. This is the set of all y such that
y = Ax; it is the span of the columns of A,
Span(a_{1}, a_{2}, ...,
a_{n}).
 Dimensions of subspaces
 Nullity. nullity(A) := dim(null space(A)).
 Rank. rank(A) := dim(row space(A)) = dim(column space(A)).
 The RankNullity Theorem: rank(A) + nullity(A) = # of columns.
 Bases. Know how to find bases for the subspaces
associated with a matrix. See my notes,
Methods for
finding bases.
Updated: 6/12/2016 (fjn)