Question

Which of the following sets are linear spaces?

a) \(\left\{X=\left(x_{1}, x_{2}, x_{3}\right)\right.\) in \(\mathbb{R}^{3}\) with the property \(\left.x_{1}-2 x_{3}=0\right\}\)

b) The set of solutions \(\vec{x}\) of \(A \vec{x}=0\), where \(A\) is an \(m \times n\) matrix.

c) The set of \(2 \times 2\) matrices \(A\) with \(\operatorname{det}(A)=0\).

d) The set of polynomials \(p(x)\) with \(\int_{-1}^{1} p(x) d x=0\).

e) The set of solutions \(y=y(t)\) of \(y^{\prime \prime}+4 y^{\prime}+y=0\).

f) The set of solutions \(y=y(t)\) of \(y^{\prime \prime}+4 y^{\prime}+y=7 e^{2 t}\).

g) Let \(S_{f}\) be the set of solutions \(u(t)\) of the differential equation \(u^{\prime \prime}-x u=f(x)\). For which continuous functions \(f\) is \(S_{f}\) a linear space? Why? [NOTE: You are not being asked to actually solve this differential equation.]