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MATH2022: Linear and Abstract Algebra
创建日期：2024-06-17 16:19:21
所属分类：数学mathematics
标签： MATH2022
Linear and Abstract Algebra

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MATH2022: Linear and Abstract Algebra 

This assignment contains four questions and is worth 5% of your total
mark. 
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.
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