Question

The vectors of the form $\left[\begin{array}{c}\mathit{a}\\ \mathit{b}\\ 0\\ \mathit{a}\end{array}\right]$(where a and b are arbitrary real numbers) forms a subspace of$R^4.$

Short answer

The above statement is true.

If W be the collection of the vectors of the form$\left[\begin{array}{c}\mathit{a}\\ \mathit{b}\\ 0\\ \mathit{a}\end{array}\right]$, then W forms a subspace of$R^4.$

Step-by-step solution

Two Step Subspace Test

• Let V be a vector space over the field K. Let W be a subset of V, then W be a subspace of V if and only if -

1. $0\in W$
2. $u,v\in W\Rightarrow u+v\in W$
3. $u\in W,\text{a}\in \text{F}\Rightarrow au\in W$

To check whether the given form vectors form a subspace of R4.

Let W be the collection of the vectors of the form $\left[\begin{array}{c}a\\ b\\ 0\\ a\end{array}\right]$in$R^4.$

Let$u,v\in W$ then we have

$u=\left[\begin{array}{c}a\\ b\\ 0\\ a\end{array}\right],v=\left[\begin{array}{c}c\\ d\\ 0\\ c\end{array}\right]$

Clearly, we see that W is non empty as

$\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]\in W$

Now,

$u+V=\left[\begin{array}{c}a\\ b\\ 0\\ a\end{array}\right]+\left[\begin{array}{c}c\\ d\\ 0\\ c\end{array}\right]=\left[\begin{array}{c}a+c\\ b+d\\ 0\\ a+c\end{array}\right]\in W$

$\Rightarrow u+v\in W$

And for $\gamma \in F,\text{and}u\in W$, we have

$\gamma u=\gamma \left[\begin{array}{c}a\\ b\\ 0\\ a\end{array}\right]\left[\begin{array}{c}\gamma a\\ \gamma b\\ 0\\ \gamma a\end{array}\right]\in W$

Final Answer

If, W be the collection of the vectors of the form $\left[\begin{array}{c}a\\ b\\ 0\\ a\end{array}\right]$in$R^4$ androle="math" localid="1664262248842" $u,v\in W$ then we have

1. $0\in W$
2. $u,v\in W\Rightarrow u+v\in W$
3. $u\in W,\text{a}\in \text{F}\Rightarrow au\in W$

Since, subset W of$R^4$ satisfies all the above three conditions, thus W is a subspace of data-custom-editor="chemistry" $R^4$.

Thus, the vectors of the form forms $\left[\begin{array}{c}a\\ b\\ 0\\ a\end{array}\right]$the subspace of$R^4$ .