Math 311-101 — Test I Review — Summer
I, 2013
General Information
Test 1 (Wednesday, June 19) will have 6 to 8 questions, some with
multiple parts. It will cover sections 1.1-1.4, 2.1-2.3, 3.1-3.4, 3.6, and
4.1 in the text. In addition, it will include material from these sets of notes:
Notes on Row Reduction;
Coordinate Vectors; 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. I will have extra office hours on Tuesday
afternoon, 11:45 am-2:30 pm, and Wednesday morning, 9-9:45 am.
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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.
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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
- No the row-reduction algorithm as outlined in my
Notes on Row Reduction. Be able to put a matrix in reduced row
echelon form.
- 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.
- 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 = x1a1 + x2
a2 + ... +xn an
where the aj'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 [A|I].
- 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.
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.
- 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: Rn,
Rm×n, Pn, C[a,b],
Ck[a,b]. (Pn is the set of polynomials of
degree less than n. So, for example, P3 is the set
of quadratics.)
- 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?
- Linear combinations. Span(v1, v2, ...,
vn). Spanning sets. Be able to
determine whether or not S is a spanning set for a vector space.
- Null space of a matrix A, N(A) = {x in
Rn | 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 =
{v1, v2, ...,
vk) is LI or LD, start with the homogeneous equation
(∗) c1v1 +
c2v2, ...,
ckvk = 0
- IF the only scalars for which the equation (∗) hold are
c1 = c2 = ... = ck = 0, then S is
LI.
- IF there are nonzero scalars for which (∗) holds are, then
S is LD.
- Basis, Dimension and Coordinates
- Definition of basis. A set S = {v1,
v2, ..., vn} is a basis for a
vector space V if and only if (i) S is LI and (ii) S spans V.
- Coordinate theorem and coordinate vectors. See my notes on
Coordinate Vectors.
- 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 Pn,
Rn, Rm×n
- Suppose that a vector space V has dim(V) = n and S = {v1,
v2, ..., vm}.
- 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
Rn | Ax = 0}.
- Row space. This is the span of all of the rows of A,
Span(r1, r2, ...,
rm).
- Column space. This is the set of all y such that
y = Ax; it is the span of the columns of A,
Span(a1, a2, ...,
an).
- Dimensions of subspaces
- Nullity. nullity(A) := dim(null space(A)).
- Rank. rank(A) := dim(row space(A)) = dim(column space(A)).
- The Rank-Nullity 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.
Linear transformations
- Definition. Know the definition of the term linear transformation.
- In cases similar to the examples in the text and ones done in class, be able to verify that a transformation is linear.
Practice tests
Updated: 6/17/2013 (fjn)