Math 307
Linear Algebra
Spring 2014
Class Notes
1.1 INTRODUCTION
Many physical quantities, such as a force, velocity, or acceleration, involve both a magnitude and
a direction. We can represent such a physical quantity mathematically by an arrow whose len
January 31, 2008
Section 1.2
Problem 15: V is not a vector space over the eld of complex numbers. Because, for example,
(1; 0; 0; :; 0) 2 V but i (1; 0; 0; :; 0) = (i; 0; 0; :; 0) does not belong to V anymore.
Problem 18: V is not a vector space because i
Math 307
Linear Algebra
Spring 2017
Exam 2 Review Problems
The following problems are intended to give you an idea as to the type of problems you might expect
to see on the exam. You should use these problems to help review general concepts. The actual ex
Homework 8 Solutions
Possible Quiz Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Homework 6 Solutions
Possible Quiz Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Homework 2 Solutions
Possible Quiz Questions (Partial solutions only)
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guidance. With this information and further work on your part, you should be
Homework 10 Solutions
Math 307: Linear Algebra
Spring 2015
Written Problems
1. Let V = C 0 ([, ]) denote the vector space over R of all continuous real-valued
functions on the interval [, ], and define the inner product
Z
1
f (t)g(t) dt
hf, gi =
on V .
Homework 5 Solutions
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guidance. With this information and further work on your part, you should be
Homework 3 Solutions
Possible Quiz Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Homework 4 Solutions
Possible Quiz Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Section 2.1
Problem 10) Suppose T : R2 ! R2 is a linear and T (1; 0) = (1; 4) and
T (1; 1) = (2; 5) : Find T (a; b) for every a; b 2 R and then nd T (2; 3) : Is T is
1-1?
Solution) Note that (a; b) = (a
T (a; b)
=
=
=
=
=
b) (1; 0) + b (1; 1) : Therefore
Section 2.3
Problem 9) We can let U (a; b) = (b; 0) and T (a; b) = (a; 0) : Then
U (T (a; b)
= U (a; 0)
= (0; 0)
But
T (U (a; b) = T (b; 0)
= (b; 0)
6= (0; 0) (if b 6= 0)
If we let
be the standard basis for F 2 and we let A = [U ] =
0
0
1
0
1 0
; then you
Homework 7 Solutions
Possible Quiz Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Solutions to problems related to 6.1
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be able to fill in the details to
ma
Math 307
Linear Algebra
Homework Solutions
5.1 EIGENVALUES AND EIGENVECTORS
Spring 2017
3. Prove Theorem 5.4: Let T be a linear operator on a vector space V , and let be an eigenvalue
of T . For all vectors ~v 2 V , ~v is an eigenvector of T corresponding
Math 307
Linear Algebra
Homework Solutions
7.2 SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
Spring 2017
1. Solve the system of dierential equations ~y 0 = A~y , where
1 3
(a) A =
3 1
The characteristic polynomial is
1 t
3
PA (t) = det
= (1 t)2 9 = t2
3
1 t
2t
Math 307
Linear Algebra
Spring 2017
Homework Solutions
2.3 COMPOSITION OF LINEAR TRANSFORMATIONS AND MATRIX MULTIPLICATION
4. Let V = cfw_(a1 , a2 ) | a1 , a2 2 R. Define operations on V as follows. For all ~u = (a1 , a2 ) and ~v = (b1 , b2 )
in V and for
Math 307
Linear Algebra
Spring 2017
Homework Solutions
6.2 GRAM-SCHMIDT ORTHOGONALIZATION PROCESS AND ORTHOGONAL COMPLEMENTS
3
1. Let V = R with the standard inner product and let W be the subspace
W = cfw_(a, b, 0) | a, b 2 R.
(c) Let ~v = (1, 2, 3) 2 V
Math 307
Homework Solutions
1.2 VECTOR SPACES
Linear Algebra
Spring 2017
1. Disprove the following statements by finding a counterexample.
(a) For all a, b 2 F and for each ~x 2 V , a~x = b~x implies a = b. (You need to first choose a specific
vector spac
Math 307
Linear Algebra
Homework Solutions
1.5 LINEAR DEPENDENCE AND LINEAR INDEPENDENCE
Spring 2017
3. Prove that the following statements are false.
(a) For all subsets S of V , if S is a linearly dependent set, then each vector in S is a linear combina
Math 307
Linear Algebra
Spring 2017
Exam 1 Review Problems
The following problems are intended to give you an idea as to the type of problems you might expect
to see on the exam. You should use these problems to help review general concepts. The actual ex
Math 307
Linear Algebra
Homework Solutions
6.3 THE ADJOINT OF A LINEAR OPERATOR
1. Find the least-squares solution to the system of
0
4x
= 2
4
2y = 0
A = @0
1
x + y = 11
Spring 2017
equations
1
0
x
2A,
~x =
,
y
1
0
1
2
~b = @ 0 A .
11
(a) Using the orth
Homework 11 Solutions
Possible Exam Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Homework 1 Solutions
Possible Quiz Questions (Partial solutions only)
The solutions for these problems are only partial; they are often written informally and are meant to offer
guidance. With this information and further work on your part, you should be
Section 1.4
Problem 4-d): The rst vector could be expressed as a linear combination
of the other two vectors if we could nd scalars a and b such that:
x3 + x2 + 2x + 13
= a 2x3 3x2 + 4x + 1 + b x3 x2 + 2x + 3
or
x3 + x2 + 2x + 13 = (2a + b) x3 + ( 3a b) x
Section 2.1
Related to Problem 5) For T : P2 (R) ! P3 (R) dened by T (f (x) =
x2 f 0 (x) + f (x), show that T is a linear map, and nd bases for N (T ) and
R (T ) :
Solution:
T is linear:
We need to show that T (f (x) + g (x) = T (f (x) + T (g (x) for ever
Section 1.6
Problem 5) Is f(1; 4; 6) ; (1; 5; 8) ; (2; 1; 1) ; (0; 1; 0)g a linearly independent
subset of R3 ?
Solution:
No, because there are more vectors in the set than the dimension of R3 :
(Note that by theorem, any linearly independent subset of R3
MATH 307 HW1
DR. T. MURPHY
(1) Determine whether the vectors emenating from the origin and terminating
at the following pairs of points are parallel.
(a) (3, 1, 2) and (6, 4, 2).
(b) (3, 1, 7) and (9, 3, 21).
(c) (5, 6, 7) and (5, 6, 7).
(d) (2, 0, 5) and
MATH 307 HW2
DR. T. MURPHY
(1) We have to show U is a subspace of W . We apply the subspace theorem.
Firstly, it is clear 0W , the zero element of W , is in U . This is because
OW = OV (since V is a subspace of W , by a theorem from class) and
similarly O