THE UNIVERSITY OF SYDNEY
PURE MATHEMATICS
MATH2061 Linear Mathematics 2024
Assignment
1. Let A =
1 1 12 −1 1
4 1 µ
 , µ ∈ R.
a) (4 marks) For each µ ∈ R find Cartesian equations for Col(A) and Null(A).
b) (2 marks) Are the columns of A linearly independent? Give full reasons for your answer
(discussion in terms of µ ∈ R, based on the definition of linear independence).
2. (3 marks) Let r ∈ R and
u =
(
1
r−r
)
, v =
(
0−4
1
)
, w =
(
3
2
r
)
.
For which values r ∈ R is the set {u,v,w} linearly independent?
3. Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, p2, p3 ∈ P2 be
given by p1(x) = 3x, p2(x) = 2x+ x2, and p3(x) = β + αx2.
a) (4 marks) Find the condition on α, β ∈ R that ensures that {p1, p2, p3} is a basis for P2.
(You are free to assume that the polynomials 1, x and x2 are linearly independent.)
b) (2 marks) In the case β = 1, write the polynomial p(x) = 1 − x − 1
2
x2 as a linear
combination of p1, p2 and p3.
4. (5 marks) Let P3 be the vector space of all real polynomials of degree at most 3. Let
S1 = {p ∈ P3 : p(x) = ax3 + bx, a, b ∈ R}, S2 = {p ∈ P3 : p(x) = bx3 + bx2, b ∈ R} and
S3 = {p ∈ P3 : p(x) = ax2 + b, a, b ∈ R, b ≥ 0}.
Determine whether S1, S2 or S3 are vector subspaces of P3, giving full reasons for your answers.
For those who are vector subspaces determine their dimension.
Math 2061: Assignment L.P. 25/1/2024