Show that the set $W$ of all vectors that are of the form $x=(a,0,a,0)$ is subspace of $R^4$.

Question

The task is: Task

Show that the set W of all vectors that are of the form x=(a,0,a,0) is subspace of R^4 by finding a set of spanning vectors of W. What is the geometric form?

My approach was to pick values for a that belongs to R and make 4 vectors. Let v1=(1,0,1,0),v2=(2,0,2,0),v3=(3,0,3,0) and v4=(4,0,4,0). Then I made a matrix of them:

1 2 3 4
0 0 0 0
1 2 3 4
0 0 0 0

I used RREF on that matrix, and got the span (-2,0,0,0),(0,-3,0,0), (0,0,0,-4).

Did I do the task correctly. And how can I know which geometric shape is that?

Answer 1

$W=R^2$ is a vector space and being a subset of $R^4$ it is a subspace. In terms of basis; $\{(1,0,0,0), (0,0,1,0)\}$ is the standard basis of $W$ and this basis is contained in the standard basis of $R^4$, and thus this way also you can conclude that $W$ is a subspace of $R^4$.