Consider the subspaces U = span{2 + x, 1 − x^2}, and W = span{−1 + x + x^2 , −x + x^2} of P2(R). Find U ∩ W

Question

The question:

Consider the subspaces U = span{2 + x, 1 − x^2}, and W = span{−1 + x + x^2, −x + x^2} of P2(R).

a) Find U ∩ W:

Theres no examples in my textbook of how to solve any of this so I have no clue what I'm doing. By reading around heres what I have so far:

Let a,b,c,d ∈ R. a(2 + x) + b(1 − x^2) = c(−1 + x + x^2) + d(−x + x^2)

we can simplify this to be:

(-b)x^2 + (a)x + (2a+b) = (c+d)x^2 + (c-d)x + (-c)

we can see that: -b = c + d, a = c - d, 2a + b = -c

I eventually found b,c,d in terms of a:

b = -5a, c = 3a, d = 2a

What do I do now? How do I show U ∩ W?

Answer 1

If your computation is correct, i.e. $b = -5a, c = 3a, d = 2a$ holds, then you simply insert the result in the description of $U$ or $W$. That is, using the description of $U$, the intersection $U \cap W$ can be expressed as the set $$ \{ a(2 + x) + -5a(1 − x^2) | a \in \mathbb{R} \} = \{ a(5x^2 + x - 3) | a \in \mathbb{R} \} $$ or, equivalently, using the description of $W$ $$ \{ c(−1 + x + x^2) + d(−x + x^2) \} = \{ 3a(−1 + x + x^2) + 2a(−x + x^2) | a \in \mathbb{R} \} $$ which is again the same descrption as the one given above, i.e. $$ \{ a (5x^2 + x - 3 ) | a \in \mathbb{R} \} \ . $$

Answer 2

You have shown that any element in $U∩W$ if and only if is of form $5ax^2+ax-3a$. You could conclude that $U∩W=\{5ax^2+ax-3a:a\in\mathbb{R}\}$