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Math 3130: Introduction to Linear Algebra,
Spring 2013
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Homework
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Most or all sections of the book have a
list of practice problems immediately preceding the exercises
for the section.
You should try to solve all these practice problems,
since they cover the most fundamental information of the section.
The solutions to the practice problems may be found
at the end of the list of exercises for the section.
The book also contains solutions to odd numbered exercises,
so these should be considered to be practice problems too. (Try them!)
Assignment
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Assigned
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Due
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Problems
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| HW1 |
1/17/13
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1/23/13
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Read Sections 1.1-1.3.
Section 1.1: 4, 12, 16, 18, 20
Section 1.2: 4, 14, 16
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| HW2 |
1/25/13
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1/30/13
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Read Sections 1.4, 2.1.
Section 1.4: 10, 12, 14
Section 2.1: 2, 6, 10, 12, 20
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| HW3 |
2/1/13
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2/6/13
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Read Sections 1.5, 1.7.
Section 1.5: 8, 20, 22
Section 1.7: 2, 10, 32
Extra:
1. Let A be the 3×3
zero matrix. Express the solution to
Ax=0 in parametric vector form.
2. Explain why, if A is an m× n matrix and m<n,
the columns of A must be dependent.
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| HW4 |
2/8/13
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2/13/13
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Read Sections 1.8-1.9.
Section 1.8: 18, 30
Section 1.9: 4, 6, 8, 10, 12, 30
Extra:
1. If p and
d are vectors in
ℝn, d≠0,
then the set of all vectors of the form
x=p+td, t∈ℝ, is a line in
ℝn.
Show that the image of this line under a linear transformation is
either a line or a point.
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| HW5 |
2/15/13
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2/20/13
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Section 1.6: 6
Section 2.1: 26, 28
Section 2.2: 2, 16, 20, 22, 24
Extra:
1. Show that if A has a left inverse, then At
has a right inverse.
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| HW6 |
2/21/13
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2/27/13
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Section 2.4: 10, 25
Section 2.5: 4, 8, 12, 20, 19, 26
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| HW7 |
3/8/13
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3/13/13
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Section 2.6: 4
Section 2.7: 4, 6
Section 2.8: 4, 6, 24
Section 2.9: 14, 18
Extra:
1. Explain why ℝ4 has no
5-dimensional subspace.
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| HW8 |
3/13/13
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3/20/13
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Section 3.1: 4, 14
Section 3.2: 4, 8, 22, 24, 34
Section 3.3: 6, 14, 22
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| HW9 |
3/20/13
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4/3/13
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Section 4.1: 22, 32
Section 4.2: 30
Section 4.3: 22
Section 4.4: 14, 16, 32
Extra:
1. Find a basis for the kernel of the linear transformation
T:P2(t)→ℝ2:
p(t)↦[p(0) p'(0)]T.
(The kernel of a linear transformation
T is the set of all x such that
T(x)=0. Observe that any correct answer to this
problem will be a set of polynomials.)
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| HW10 |
4/3/13
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4/10/13
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Section 4.5: 32
Section 4.6: 4, 8, 30
Section 4.7: 6, 8
Section 4.9: 2
Chapter 4 Supplementary Exercises: 4, 10, 12
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| HW11 |
4/11/13
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4/17/13
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Section 5.1: 8, 16, 22, 30
Section 5.2: 10, 20
Section 5.3: 8, 12
Extra:
1. Show that if λ is an e-value for A and k is an integer,
then λk is an e-value for
Ak. (Don't neglect the case where k is a negative
integer if A is invertible.)
2. Suppose that A is a block upper triangular matrix with diagonal blocks
A1, …, Ak. Show that the characteristic
polynomial of A is the product of the characteristic polynomials
of the Ai's.
3. Find the e-values, e-spaces and characteristic polynomial
of the n×n matrix whose entries all equal 1.
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| HW12 |
4/17/13
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4/24/13
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Section 5.4: 14, 19, 20, 22
Section 5.5: 2
Extra:
1. Suppose that A is a 2×2-matrix and that
det(A-I) = -16 and det(A-2I) = -15. What are the e-values of A?
2. Let
Pn(t) be the space of
polynomials of degree at most n in the variable t.
Let D:P3(t)→P3(t):
f(t)↦f'(t) be the operation of differentiation.
Find the characteristic polynomial, e-values, and
e-spaces of D.
3. Let T:M2×2(ℝ)→M2×2(ℝ):
M↦MT be the operation of transpose.
Find the characteristic polynomial, e-values, and
e-spaces of T.
4.
Let E(t) be the subspace of Pn(t)
consisting of polynomials
whose monomials have even exponent (like 1, t2, t4, etc)
and let O(t)
be the subspace consisting of polynomials
whose monomials have odd exponent
(like t, t3, etc). Show that
Pn(t) = E(t) ⊕ O(t).
What are the dimensions of
Pn(t), E(t) and O(t) when n=100?
5. Suppose that T:V→V is a linear transformation
defined on a finite dimensional vector space V and that
T has an eigenvalue λ whose associated
eigenspace is all of V (i.e. Vλ = V).
What are the possible matrices B[T]B
for different bases B?
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