Question

Although problems are categorized by topics, this should not be taken very seriously since
many problems fit equally well in several different topics.
Notation: We occasionally write M(n, F) for the ring of all n × n matrices over the field F, where
F is either R or C.
Basic Definitions
1. Which of the following sets are linear spaces?
a) {X = (x1, x2, x3) in R
3
with the property x1 − 2x3 = 0}
b) The set of solutions x of Ax = 0, where A is an m × n matrix.
c) The set of 2 × 2 matrices A with det(A) = 0.
d) The set of polynomials p(x) with
R
1
−1
p(x) dx = 0.
e) The set of solutions y = y(t) of y

Answer 1

a, b, and d, assuming you mean the riemann integral in the interval [-1, 1]