Question

Question: For any vector $\mathit{v}\mathbf{\in }\mathit{V}$and any element $\mathit{a}\mathbf{\in }\mathit{F}$, prove that

(a)${\mathit{O}}_{\mathbf{F}\mathbf{V}}\mathbf{=}{\mathit{O}}_{\mathbf{V}}$

(b)$\mathit{a}{\mathbf{0}}_{\mathbf{v}}\mathbf{=}{\mathbf{0}}_{\mathbf{v}}$

(c)$\mathbf{-}\mathbf{\left(}\mathbf{av}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{a}\mathbf{)}\mathit{V}\mathbf{=}\mathit{a}\mathbf{(}\mathbf{-}\mathbf{a}\mathbf{)}$

Short answer

Answer:

(a)${\mathit{O}}_{\mathbf{F}\mathbf{V}}\mathbf{=}\mathbf{0}\mathit{V}$

(b)$\mathit{a}{\mathbf{0}}_{\mathbf{v}}\mathbf{=}{\mathbf{0}}_{\mathbf{v}}$

(c)$\mathbf{-}\mathbf{\left(}\mathbf{av}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{a}\mathbf{)}\mathit{V}\mathbf{=}\mathit{a}\mathbf{(}\mathbf{-}\mathbf{a}\mathbf{)}$

Step-by-step solution

Definition of space.

A space consisting of vector, together with the associative operations of addition of vectors and the associative and distributive operation of multiplication of vectors by scalars.

Step 2:Showing that OFV=0V

Let V be any vector space over a field.

Let $v\in V$and$a\in F$.

(a)

Let $0v=\left(0+0\right)v$

As $0v\in V$and V is an additive group.

Therefore by cancellation law it follows that

$0v=0$

Therefore, $0v=0\in V$

Step-3: Showing that a0v=0v

(b)

Let

$$\begin{gathered} a{0}_v=a\left(0_v+0_v\right) \\ =a{0}_v+a{0}_v \end{gathered}$$

This gives

$a{0}_v+0_v=a{0}_v+a{0}_v$

Therefore by cancellation laws it follow that,

$0_v=a{0}_v$

This complete the claim.

Step-4: Showing that -(av)=(-a)v=a(-v).

(c

As

$$\begin{gathered} av+\left(-a\right)v=\left(a+\left(-a\right)\right)v \\ =0v \end{gathered}$$

Therefore,

$\left(-a\right)v=-\left(av\right)$ …(i)

Similarly,

$$\begin{gathered} av+\left(-a\right)v=\left(a+\left(-a\right)\right)v \\ =a0 \end{gathered}$$

As $a.0=0$

So,

$a\left(-v\right)=-\left(av\right)$ …(ii)

From (i) and (ii)

$$\begin{gathered} \left(-a\right)v=a\left(-v\right) \\ =-\left(av\right) \end{gathered}$$

Hence proved.