The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH2022: Linear and Abstract Algebra Semester 1, 2024
Lecturer: Leah Neves and Sam Jeralds
Due 11:59pm Sunday 12 May 2024.
This assignment contains four questions and is worth 5% of your total
mark. It must be uploaded through the MATH2022 Canvas page
https://canvas.sydney.edu.au/courses/56888/assignments/518806.
Please include your SID but not your name, as anonymous marking will be
implemented.
1. For each of the vector spaces below with specified bases B, find the coordinate vector [v]B
of the given vector v with respect to B.
(a) V = R3, B = {(1, 0, 3), (2, 1, 8), (1,−1, 2)}, v = (3,−5, 4).
(b) V = P2(R), B = {1 + t2, t + t2, 1 + 2t + t2}, v = 1 + 4t + 7t2.
2. Consider the matrix M over Z7, where
M =

1 1 6 2 6
4 1 4 2 5
5 2 3 5 0
3 4 6 2 4
1 2 1 4 3
 .
(a) Compute rank(M) and nullity(M) over Z7.
(b) Find a basis of the null space Nul(M) over Z7.
3. Let V be a vector space and X = {x1,x2, . . . ,xk} a set of linearly independent vectors in
V . Prove that any nonempty subset Y ⊆ X is also a linearly independent set of vectors
in V .
4. Throughout this problem, let V be the vector space of polynomials in two variables x, y
over R with degree at most two. That is,
V = {a0 + a1x + a2y + a3x2 + a4xy + a5y2 : ai ∈ R}.
This is a six-dimensional vector space with basis B = {1, x, y, x2, xy, y2}.
(a) Let S be the “variable swap” on a polynomial: it replaces x with y, and y with x.
For example, S(3+2x−4xy+y2) = 3+2y−4xy+x2. For each of the polynomials
p(x, y) below, compute S(p(x, y)).
(i) p(x, y) = 3− 4x + xy + 2y2
(ii) p(x, y) = x2 − 4x + 3xy − 4y + y2 − 1
(b) A polynomial p(x, y) is called symmetric if S(p(x, y)) = p(x, y). Define
VS := {p(x, y) ∈ V : p(x, y) is symmetric} ⊂ V.
Copyright© 2024 The University of Sydney 1
Show that VS is a subspace by proving it is nonempty and closed under addition
and scalar multiplication.
(c) Find the dimension of VS and a basis of VS as follows: Take a general polynomial
p(x, y) = a0 + a1x + a2y + a3x
2 + a4xy + a5y
2 in V . Determine the relationships
the coefficients must satisfy to force S(p(x, y)) = p(x, y). Grouping together terms
with the same coefficients will reveal the basis elements. (Technically this just
shows spanning, but linear independence also holds).
Remark: The set VS is called the space of symmetric polynomials in two variables of
degree at most two. Such polynomials and their generalizations play a major role in the
study of algebra, in particular in algebraic combinatorics and representation theory.