Linear Algebra – 4 Questions – 23 parts

MAT10720 Linear Algebra

Assignment 3 Session 2 in 2013

Assessing content from topics 5-9

Q1 Vector Basics

a. Determine whether the vectors v1 = (1,2,3), v2 = (-1,2,-1) and v3 = (-1,1,3) form an

orthogonal set. 3 marks

b. Calculate the norm of v = (8,-6) and find the unit vector in the same direction as v.

2 marks

c. Determine whether the points P1 = (-1,2,2), P2 = (3,3,4), P3 = (2,-2,10) and P4 = (0,2,2)

lie on the same plane using the scalar triple product. 4 marks

d. Determine the angle in radians between u1 = (2,4,-5) and u2 = (-1,3,-2). 3 marks

e. Find a vector parallel to u = (4,-9,3,0,8) in R
5
. 1 mark

f. Calculate the cross product of v1 = (2,1,-3), v2 = (0,4,5). What is the angle between the

resulting vector and v1, and the resulting vector and v2? 3 marks

g. Find the vector component of u along a and the vector component of u orthogonal to a

for u = (7,-2,3) and a = (-8,-4,-4). 3 marks

h. Calculate the coordinates of P4 so that the quadrilateral formed by the vertices

P1 = (−1, −2), P2 = (4, −1), P3 = (5, 2) and P4 is a parallelogram. 4 marks

Q2 Vector Spaces and Subspaces

a. Determine if the set of all triples of real numbers of the form (0,y,z) where y=z with

standard operations on R
3

is a vector space. 8 marks

b. Determine the following:

i. If the set of all matrices of the form

0

where 0
0

a
a

b

b

 
+ = 

 
is a subspace of M22.

3 marks

ii. If the set of vectors with integer components is a subspace of R
n
. 3 marks

MAT10720 Linear Algebra Assignment 3 Session 2 in 2013

Page 2 of 3

Q3 Linear Independence and Spanning Sets

a. Can the vector w = (2,1,7) be expressed as a linear combination of the vectors u = (1,-1,2)

and v = (-3,2,-7)? If so, give the dependency equation. Comment on whether a set of all

three vectors would be linearly dependent or independent. 3 marks

b. Calculate the value of c (if possible) for u1 = (5, 3 − c) and u2 = (c + 9, 3c + 1) to be

linearly dependent. 3 marks

c. Determine if vectors v1 = (1,2,4), v2 = (0,3,8) and v3 = (5,1,8) span R
3
. Comment on

whether the set of the three vectors would be linearly dependent or independent.

6 marks

Q4 Bases, Row, Column and Null Space

a. For the following linear system:

1 2 3 4 5

1 2 3 4 5

1 3 4 5

1 2 3 4 5

2 8 3 10 0

0

3 9 0

2 4 2 4 0

x

x x x x

x x x x x

x x x x
x x x x x

− − + − =

+ − + − =

− − − =

+ − − + =

i. Represent this linear system in the form Ax=b. 1 mark

ii. Explain what the null space of the coefficient matrix A is in terms of the linear

system. 1 mark

iii. Find a basis for the null space of A. 8 marks

iv. Find the rank and nullity of matrix A. 2 marks

b. For the Bases B = {u1, u2}, where u1 = (2, 1), u2 = (1, 4), and B’ = {v1, v2}, where

v1 = (3, 1) and v2 = (-2, 1):

i. What is the effect of multiplying a vector by the transition matrix PB’→B? 1 mark

ii. Find the transition matrix PB’→B. 4 marks

iii. Using the result from (ii), find the transition matrix PB→B’. 2 marks

iv. If the coordinate vector w relative to basis B’is
2

1

 
 
 

, calculate the coordinate vector

of w relative to B. 1 mark

MAT10720 Linear Algebra Assignment 3 Session 2 in 2013

Page 3 of 3

c. Find a subset of the vectors given below that forms a basis for the space spanned by

these vectors: v1 = (2,1,1,-2), v2 = (-4,-2,-2,4) v3 = (3,-2,1,0), v4 = (0,14,2,-12), and

v5 = (1,-4,1,1) 6 marks

END OF ASSIGNMENT 3