School of Mathematics and Statistics University of New South Wales
V T C J
M12 T11 T14 W12
M13 T12 T15 W15
M15
MATH2501 Linear Algebra
SESSION 2, 2019 TEST 2 Version A
Student’s Surname Initials Student Number
Questions: 4 Pages: 2 Total marks: 18 Time allowed: 40 minutes
Q1 [5 marks] Find the projection of e1 ∈ R4 onto the subspace
W = span


1
1
1
−1
 ,

1
−1
1
1

 .
Q2 [5 marks] Find QR factorisation of the matrix
A =
(
5 17
12 7
)
.
Q3 [3 marks] Let V = (V,+, ·,R) and W = (W,+, ·,R) be vector spaces and let T : V → W be a
linear map.
a) Give definition of the null space nullT of the map T .
b) Let V = P2(R) and W = R2. Consider the subspace (you do not have to proof that it is a
subspace).
V =
{
p ∈ P2(R) : p(1) = 0 and p(−1) = 0
}
⊆ P2(R).
Find a linear map T such that
V = nullT.
You do not have to show that the map T is linear.
Q4 [5 marks] Let V = (V,+, ·,R) be a vector space and let B = {v1,v2,v3} ⊆ V be a basis.
a) Define what it means that the triple (a1, a2, a3) ∈ R3 is the coordinate vector of x ∈ V with
respect to the basis B.
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School of Mathematics and Statistics University of New South Wales
b) Find the coordinate vector of v1 with respect to the basis B.
c) Define what it means that a matrix A ∈M3,3(R) is the matrix of the linear map T : V → V
with respect to the basis B.
d) Let
A =
0 1 01 0 0
0 0 0
 .
be the matrix of the map T with respect to the basis B. Find T (v1) in terms of B.
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