## Acalytica

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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.]
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