Course
2B Linear Algebra 2014
Student Responsibilities
The information contained on this sheet may change! It is your
responsibility as a student to check regularily for updates this courses
Moodle site and your student email.
Lectures
In this course we w
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True/False
a) If S and S are two linearly independent sets of vectors spanning
the same subspace W of some vector space V then S = S .
b) A set of vectors containing a single non-zero vector is linearly
independent.
c) If v is a non-zero vector in a vecto
Ex Sheet
8
2B Linear Algebra 2014
True/False
a) A vector is always a line segment with direction.
b) A matrix can be a vector.
c) A function can be a vector.
d) The set Mmn (R) is a vector space only if m = n.
e) The set of all polynomials over R of degre
Ex Sheet
6
2B Linear Algebra 2014
True/False
a) If is an eigenvalue of a matrix A then Ax + x = 0 for some
x = 0.
b) Suppose a matrix A has eigenvalue 1. Then the 1-eigenspace of A
consists of all vectors x so that Ax = x.
c) An eigenvalue can equal 0 but
Ex Sheet
3
2B Linear Algebra 2014
1
.
True/False
1
a) If A, B and C are n n matrices and AB = AC then B = C.
b) If A and B are n n matrices and AB = I then B = A1 .
c) If A is an invertible matrix then the inverse of A can be written
1
as A .
d) If A is i
' 0835186203
University School of Mathematics
Maths 2B Feedback Exercise
qulasgow & Statistics
cover Sheet Deadline: Wednesday 12th November 3pm
Student Details:
Student Number TU Surname
Marker Only:
I D {11 [131
Write Answer Below (continue on b
r SOL 0710M? .1
. University School 91 Mathematics
Maths 2B Feedback Exerc1se Ema qxlasgow & Statistics
cover Sheet Deadline: Wednesday 26th November 3pm
Student Details:
1 Student Number TU Surname
Marker Only:
\ Um [1:13 I
Write Answer Below (cont
Someday, ?th December, 2012
Up to you
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2B - Linear Algebra
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphical disp
True/False
a) For all c, d R, the map T : R R that assigns to each x R the
real number cx + d is linear.
b) For all m n real matrices A then the map T : Rn Rm given by
T (v) = Av is linear.
c) Any linear transformation is completely determined by the imag
Someday, ?th December, 2013
Up to you
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2B - Linear Algebra
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphical disp
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80 L U (mi(e ouiga Im!
r 0835186203 E UniVI'SitY] School of Mathematics |
Maths 2B Feedback Exercise qf Glasgow & Statistics
Cover Sheet
Deadline: Wednesday 8th October 3pm
Student Details:
Student Number TU Surname
EEDJIDDJ
Mark
Ex Sheet
2
2B Linear Algebra 2014
1
.
True/False
1
a) For a matrix A the (i, j)-entry of A in the entry in column i and
row j.
b) Consider the matrix
0
C = 0
3
0
7
0
2
0
0
Then C is diagonal with diagonal entries 2, 7 and 3.
c) Consider the matrix
4
D = 0
Ex Sheet
1
2B Linear Algebra 2014
1
.
True/False
1
a) The vector
vectors
1
1
can be written as a linear combination of the
1
4
and
.
0
0
b) A system of linear equations with augmented matrix [ A|b] has
a unique solution if and only if b is a linear combin
Ex Sheet
0
2B Linear Algebra 2014
This exercise sheet revises material from Level 1 that is assumed
knowledge for 2B. Before the rst lecture, you should remind yourself
of these topics by reading your Level 1 notes and/or the textbook
sections indicated,
True/False
a) Any non-zero vector v Rn is an eigenvector of the identity I.
b) If v is an eigenvector of A Mnn (R) corresponding to , then
v is an eigenvector of A corresponding to .
c) If v and w are eigenvectors of A Mnn (R) corresponding to ,
then v +
True/False
a) If B is a basis for R2 then the matrix whose columns are the vectors
in B must have determinant 0.
b) Let A be an n n matrix. If nullity( A) = 0 then the columns of A
are linearly dependent.
c) Let A be an n n matrix. Then rank( A) = n if an
Course
2B Linear Algebra 2014
i
.
Advice for Degree Exam
The exam this year is quite a lot more theoretical than those
before 2012.1 See the sections below on Statements of Denitions
and on Proofs and Theory for further explanation of what you are
expecte
Ex Sheet
6
2B Linear Algebra 2014
True/False
a) If is an eigenvalue of a matrix A then Ax + x = 0 for some
x = 0.
b) Suppose a matrix A has eigenvalue 1. Then the 1-eigenspace of A
consists of all vectors x so that Ax = x.
c) An eigenvalue can equal 0 but
True/False
a) If S and S are two linearly independent sets of vectors spanning
the same subspace W of some vector space V then S = S .
b) A set of vectors containing a single non-zero vector is linearly
independent.
c) If v is a non-zero vector in a vecto
True/False
a) If B is a basis for R2 then the matrix whose columns are the vectors
in B must have determinant 0.
b) Let A be an n n matrix. If nullity( A) = 0 then the columns of A
are linearly dependent.
c) Let A be an n n matrix. Then rank( A) = n if an
This exercise sheet revises material from Level 1 that is assumed
knowledge for 2B. Before the rst lecture, you should remind yourself
of these topics by reading your Level 1 notes and/or the textbook
sections indicated, doing these exercises and doing We
True/False
a) Any non-zero vector v Rn is an eigenvector of the identity I.
b) If v is an eigenvector of A Mnn (R) corresponding to , then
v is an eigenvector of A corresponding to .
c) If v and w are eigenvectors of A Mnn (R) corresponding to ,
then v +
Ex Sheet
8
2B Linear Algebra 2014
True/False
a) A vector is always a line segment with direction.
b) A matrix can be a vector.
c) A function can be a vector.
d) The set Mmn (R) is a vector space only if m = n.
e) The set of all polynomials over R of degre
Ex S
X
.
2B Linear Algebra 2014
Feedback Exercise 2
These questions relate to the material in Lectures 69 and Exercise
Sheets 34.
a) Let
0 1
3 0
6 0
0
A = 0
1
2014
and b = 3000 .
0
i) Use the GaussJordan method to nd A1 .
ii) Use your answer to (i) to nd
Ex Sheet
2
2B Linear Algebra 2014
1
.
True/False
1
a) For a matrix A the (i, j)-entry of A in the entry in column i and
row j.
b) Consider the matrix
0
C = 0
3
0
7
0
2
0
0
Then C is diagonal with diagonal entries 2, 7 and 3.
c) Consider the matrix
4
D = 0