Change of coordinates matrix, proper subspace I don’t fully understand this exercise and it’s really frustrating.
It says something like this:

Consider the space $P_2[R]$ with basis: 
$B_1 = \{x^2 + x + 1, x, x - 1\}$
$B_2 = \{x^2 - x + 1, x, 2\}$
If $S \in P_2$ is a “proper” subspace and $L_1$ and $L_2$ bases of $S$.
a) Does the change or coordinates matrix from $L_1$ to $B_1$ exist? What would be their size?
b) Does the change or coordinates matrix from $L_1$ to $L_1$ exist? What would be their size?

I did this:
a) If $S$ is a proper subspace of $P_2$ then its basis will have fewer vectors than $P_2$, so $S$ will have one or two basis vectors, right?
I have tried to get the bases of $S$ without success. I tried to write a linear combination. I think that they will be vectors in $P_2$.
I did this:

$L_1\colon ax^2 + bx + c = \alpha(x^2 + x + 1) + \beta(x) + \delta(x - 1)$
$L_2\colon ax^2 + bx + c = \alpha(x^2 - x + 1) + \beta(x) + \delta(2)$

But I don’t know what to do now. How can I get the bases of $S$ with the given information?
Anyway, I think a) is false because $B_1$ has 3 components, I mean: $\{x^2 + x + 1, x, x - 1\}$ and $S$ is a proper subspace. Then $L_1$ and $L_2$ will have 1 or 2 elements, am I right?
I think b) is true but I’m not sure and I don’t know how to prove it.

a)Does the change or coordinates matrix from L₁ to B₁ exist?What would be their size?
It can't exist because $L_1$ has $1$ or $2$ elements whereas $B_1$ has $3$ elements. The matrix of the change of basis should be invertible.
b)Does the change or coordinates matrix from L₁ to L₁ exist?What would be their size?
Of course it exist and the size id d-by-d with d dimension of S.
