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