Instructions:
This is not a comprehensive review: there are concepts you need to know that
are
not included. Be sure you study all the sections of the book. You should know
correct statements
of definitions and important theorems, be able to state concepts in your own
terms, and apply
procedures.
For full credit, you must show work or give some explanation of how you
reached each answer. You
must communicate to me how you reached your answer.
1. Matrix Operations. Know how to add , subtract , and multiply matrices, and
how to multiply
matrices by a scalar. Know the interpretations of matrix multiplication.
a. When you multiply matrices A and B then each column of AB is a linear
combination of the
columns of what matrix? Why?
b. If A is an m x n matrix, and B is p
x q, under what condition is the product AB defined,
and
what would then be the dimensions?
c. If A, B, C are all 5 x 5 matrices, then is the
fol lowing statement something that must hold,
might hold, or cannot hold?
A(B + C) = AB + AC
d. If A, B are n x n matrices, then is the
following statement something that must hold, might
hold, or cannot hold?
(A + B)(A - B) = A^2 - B^2
e. If A, B, C are all 3 x 3 matrices, and if AB =
AC, then can you cancel the A on each side to
get B = C? Explain.
2. Inverse Matrices
a. Give a definition for inverse matrix
b. Explain how to find the inverse of a matrix, or to tell that n one exists
c. Circle all of the statements below which ARE equivalent to invertibility
of an n x n matrix A
• The rref of A is an identity matrix
• The equation Ax = b has free variables for all b .
• Every column of A is a pivot column.
• The columns of A are linearly independent
• The rank of A is n
• The de terminant of A is 1.
• The homogeneous equation for A has a nontrivial solution .
3. Elementary Matrices. Write a one page essay explaining what elementary
matrices are, how we
know they are all invertible, and how we know that every invertible matrix is a
product of elementary
matrices.
4. Definitions [20 points]. State the definitions of the items on this and
the next page.
a. The column space of a matrix
b. A vector space V .
c. A linearly independent set of vectors in a vector space V:
d. The dimension of a vector space V .
e. The kernel of a linear transformation T defined on a vector space V:
5. Cramer's Rule
a. State Cramer's rule for solving a linear system Ax = b where A is an
invertible n x n matrix,
b is a given vector in Rn, and x is an unknown vector in Rn.
b. Use Cramer's rule to solve the system
6. Properties of Determinants : T/F Mark each statement true or false, and give
a reason
to justify your answer. (You will not get credit for any correct answer
unless you give a
justification.) For all items, A and B are assumed to be n x
n matrices.
____a. If the columns of A are independent then detA = 1.
____b. If the rows of A are dependent then detA = 0.
____c. det(AB) = det(BA).
7.Computing determinants. Compute the determinant of each of the following,
showing
your work or stating a property of determinants that justifies your answer.
8.Subspaces.
a. It is known that the set of all 4x4 matrices
forms a vector space. Is the set of invertible 4x4
matrices a subspace? Explain.
b. In R^2 consider H = {(x ,y)|x
≠ y}, the set of all vectors with unequal entries. Is
this set closed
under addition? Explain.
9.Coordinate Vectors and Change of Basis. For both parts of this problem, let P2 be
the space of polynomials of degree two or less . The set B = {1, t + 1, (t + 1)^2}
is a basis for P2.
a. If
find
the polynomial q(t)
b. Find the coordinate vector
for the polynomial p(t) = 3t + 5
10.Special subspaces. For this problem, use the following matrices:
The matrix B was obtained by performing row operations on the matrix A: Using
this information,
find a basis for …
a. the column space of A
b. the row space of A
c. the null space of A
11.Basis and dimension results. Circle the correct response to complete each
statement
below, and ALSO state a theorem or give a reason that justifies your
choice.
a. If V is a 7 dimensional vector space and S is a set of 10 vectors, then
the elements of S _____
be linearly independent.
Circle one: must might cannot
Justifying Theorem or Reason:
b. If V is a 4 dimensional vector space, and if the vectors u, v, w,
and x are linearly independent,
then they ____ be a spanning set for V:
Circle one: must might cannot.
Justifying Theorem or Reason:
c. In a vector space V, if the vectors a, b, and c span V , then the
dimension of V must be ____ 3.
Circle one: < ≤ = ≥ >
Justifying Theorem or Reason:
12.Matrix for a linear transformation.
A sequence a = (a0, a1, a2, …) is Fibonacci- like if
each term after a1 is the sum of the two preceding
terms. That is, a2 = a1+a0, a3 = a2+a1, a4 = a3+a2, etc. The set of all such
sequences is a vector
space, denoted in this problem by V: The standard basis for V is
The left shift operation,
is defined for elements of V as follows. If
then
.
That is,
has the effect of erasing the first term of any sequence. This is a
linear transformation from V to itself.
Your task: find the matrix that represents
with respect to the basis B: (This is the matrix A for
which the following equation holds:
