Let $V$ be a nonempty set on which are defined operations "$+$" (called addition) and "$\cdot$" (called scalar multiplication).

$V$ is a vector space (over $\BF$) if the following hold for all $\bfx,\bfy,\bfz\in V$ and all $\alpha,\beta\in\BF$:

(V1)	$\bfx+\bfy\in V$ (closure)
(V2)	$\bfx+\bfy=\bfy+\bfx$ (additive commutativity)
(V3)	$\bfx+(\bfy+\bfz)=(\bfx+\bfy)+\bfz$ (additive associativity)

(V1)	$\bfx+\bfv\in V$ (closure)
(V2)	$\bfx+\bfy=\bfy+\bfx$ (additive commutativity)
(V3)	$\bfx+(\bfy+\bfz)=(\bfx+\bfy)+\bfz$ (additive associativity)
(V4)	$\exists\,{\bf 0}\in V$ such that $\bfx+{\bf 0}=\bfx$ (zero vector, or additive identity)
(V5)	For each $\bfx\in V$, $\exists\,(-\bfx)\in V$ such that $\bfx+(-\bfx)={\bf 0}$ (additive inverse)
(V6)	$\alpha \bfx\in V$ (closure)
(V7)	$\alpha(\bfx+\bfy)=\alpha\bfx+\alpha\bfy$ (multiplicative-additive distributivity)
(V8)	$\left(\alpha + \beta \right)\bfx=\alpha\bfx+\beta\bfx$ (additive-multiplicative distributivity)
(V9)	$\alpha(\beta\bfx)=(\alpha\beta)\bfx$ (multiplicative-multiplicative distributivity)
(V10)	$1\bfx=\bfx$ (multiplicative identity)

1) Identify elements of the set: $$\BF^n = \big\{ \u = \left( u_1, u_2, \ldots , u_n\right)~|~ u_1, u_2, \ldots , u_n \in \BF \big\}$$

2) & 3) Check for closure of addition and scalar multiplication: $$\u+ \v = \left( u_1+v_1, u_2+v_2, \ldots , u_n+v_n\right)$$ $$\;\,k \cdot \u = \left( ku_1, ku_2, \ldots , ku_n\right)$$

4) Identify the vector zero: $\mathbf 0 = \left( 0, 0, \ldots , 0\right)$

5) Identify the inverse additive: $ - \mathbf u = \left( -u_1, -u_2, \ldots , -u_n\right)$

1) Identify elements of the set: $$\BF^n = \big\{ \u = \left( u_1, u_2, \ldots , u_n\right)~|~ u_1, u_2, \ldots , u_n \in \BF \big\}$$

2) & 3) Check for closure of addition and scalar multiplication: $$\u+ \v = \left( u_1+v_1, u_2+v_2, \ldots , u_n+v_n\right)$$ $$\;\,k \cdot \u = \left( ku_1, ku_2, \ldots , ku_n\right)$$

4) Identify the vector zero: $\mathbf 0 = \left( 0, 0, \ldots , 0\right)$

5) Identify the inverse additive: $ - \mathbf u = \left( -u_1, -u_2, \ldots , -u_n\right)$

Now you can continue checking that the other properties hold. 📝

Remark: This is just an strategy you can use. But if you prefer, you can verify each property one by one in the given order, that is, from (V1) to (V10).

1) $M_{m,n} = \left\{ \left( \begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \\ \end{array} \right) ~\Bigg|~ a_{ij}\in \BF, 1\leq i\leq m, 1\leq j\leq n \right\}$

2) & 3) Usual addition and scalar multiplication for matrices.

4) $\mathbf 0 = \left( \begin{array}{ccc} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \\ \end{array} \right)$

5) $- \mathbf u = \left( \begin{array}{ccc} -a_{11} & \cdots & -a_{1n} \\ \vdots & \ddots & \vdots \\ -a_{m1} & \cdots & -a_{mn} \\ \end{array} \right)$

1) $\mathbf f, \mathbf g\in C[a,b],$ often represented as $f(x), g(x).$

2) & 3) $\left(\,f+g\right)(x) = f(x) + g(x)$
$\qquad \quad \,\, (k \cdot f)(x) = kf(x)$

4) $\mathbf 0 = \,?$ 🤔

5) $ - \mathbf f = -f(x) $

1) $\mathbf p \in P_n(\BF)$, with $\mathbf p = a_0 + a_1x + \cdots + a_n x^n$ and $a_k\in \BF,$ $\forall k.$

2) & 3) Operations similar to Example 4.

4) $\mathbf 0 = \,?$ 🤔

5) $ - \mathbf p = -p(x) .$

For example: $y''+p(x) y' + q(x) y = 0 $ ($y=y(x)$).

1) Let $y_1$ and $y_2$ be solutions.

2) Operations similar to Example 4.

3) Superposition principle: $y_1+y_2 $ is also a solution.

4) The zero vector is the zero function.

5) Given a solution $y$, $-y$ is also a solution.