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Mmex 1

This document provides 8 exercises related to linear algebra concepts including: 1) Determining if lines defined by vector equations are parallel or coincident 2) Identifying collinear points in R3 3) Analyzing relationships between lines defined parametrically or between points 4) Computing defined matrix expressions and determining orders of matrix products 5) Proving properties of inverses of invertible matrices 6) Computing inverses of 2x2 matrices 7) Working an additional exercise on vectors The objectives are to review concepts of vectors in R2 and R3, lines defined by vectors, matrices, inverses, transposes, and matrix operations.

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76 views2 pages

Mmex 1

This document provides 8 exercises related to linear algebra concepts including: 1) Determining if lines defined by vector equations are parallel or coincident 2) Identifying collinear points in R3 3) Analyzing relationships between lines defined parametrically or between points 4) Computing defined matrix expressions and determining orders of matrix products 5) Proving properties of inverses of invertible matrices 6) Computing inverses of 2x2 matrices 7) Working an additional exercise on vectors The objectives are to review concepts of vectors in R2 and R3, lines defined by vectors, matrices, inverses, transposes, and matrix operations.

Uploaded by

Daniel Espinal
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Exercise Set 1

1. Are the lines with equations

x
y
z

1
2
2

+ s

3
5
7

and

x
y
z

7
12
16

+ t

6
10
14

parallel or coincident?
2. Which of the following sets of points in R
3
are collinear that is, lie on a line?
(a) (2, 1, 4), (4, 4, 1), (6, 7, 6)
(b) (1, 2, 3), (4, 2, 1), (1, 1, 2).
3. Let
1
be the line with equation (x, y, z)
T
= (1, 1, 2)
T
+ t(1, 2, 3)
T
,
2
the line through
(5, 7, 4) and (8, 13, 3) and
3
the line through (1, 17, 6) parallel to the vector (7, 4, 3)
T
.
Show that
(a)
1
and
3
are skew, (b)
1
and
2
intersect (c)
2
and
3
intersect.
Find the points of intersection.
Determine whether each pair of intersecting lines is orthogonal. If not, nd the angle
between them.
4. Given the matrices:
A =

2 1
1 1
0 3

b =

1
1
1

C =

1 2 1
3 0 1
4 1 1

D =

0 1
2 5
6 3

Which of the following matrix expressions are dened? Compute those which are dened.
(a) Ab
(f) DA
T
+C
(b) CA
(g) b
T
b
(c) A+Cb
(h) bb
T
(d) A+D
(i) Cb
(e) b
T
D
5. If a and b are both column matrices of the same order n, what is the order of the
matrix product a
T
b?
What is the order of the matrix expression b
T
a?
What is the relationship between b
T
a and a
T
b?
6. Using the denition of the inverse of a matrix and the principles of matrix algebra, prove
that if A and B are invertible matrices of the same order, then AB is invertible and
(AB)
1
= B
1
A
1
.
7. (a) Let B =

2 5
0 1

. Find B
1
.
Hint. Let B
1
=

x y
z w

and solve the system of (four) equations given by


the matrix equation BB
1
= I:

2 5
0 1

x y
z w

1 0
0 1

.
(b) Show in general (by multiplying AA
1
): if A is a 2 2 matrix given by
A =

a b
c d

and if ad bc = 0 then
A
1
=
1
(ad bc)

d b
c a

.
Note. The term (ad bc) is the determinant of A, denoted Det(A) or |A|,
|A| =

a b
c d

= ad bc
8. Look through Background You Should Know and Exercises (see the rst two pages of
the Study Pack or look on the website). Work Exercise 3.3.
If there is any material with which you are completely unfamiliar, send an email with this
information to MA100@maths.lse.ac.uk with the subject missing background.
Reading: Lecture notes 1 for Calculus and Linear Algebra;
Binmore and Davies see lecture notes;
Anton Sections 1.31.4. (Review of vectors - Chapter 3).
Objectives.
(i) Know concept of vectors in R
2
and R
3
, and nd a vector equation of a line.
(ii) Know concept of a matrix, its inverse and transpose, and matrix operations.

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