# Find all vectors $x ∈ W$ that give the matrix

Let $$W$$ be a vector space with $$dim(W ) = 3$$. Find all vectors $$x ∈ W$$ such that there exists some basis α for which $$[x]_α$$ = $$\begin{bmatrix} 1 \\ 0 \\0 \end{bmatrix}$$

I'm stuck on this question here, not completely sure how to go about it. Here are the thoughts I've had so far:

$$Ax = \begin{bmatrix} 1 \\ 0 \\0 \end{bmatrix}$$. So, I'm solving for some vector $$x$$ where all $$A$$ matrices give that matrix? Could $$x$$ be $$\begin{bmatrix} x_1 \\ 0 \\0 \end{bmatrix}$$? Do I need to somehow find $$a$$, the basis to solve this?

Would greatly appreciate any help on this, thanks!

If $$x=0_W$$, then the problem has no solution, since every coordinate of nuell vector with respect to any basis is $$0$$.
And if $$x\ne0_W$$, just take two vectors $$y$$ and $$z$$ such that $$B=\{x,y,z\}$$ is linearly independent. Then $$B$$ is a basis of $$W$$ and$$x=1.x+0.y+0.z.$$
• is this around the same idea as what I wrote for $x$? Dec 4, 2021 at 19:31