# Show that a vector space is spanned by a given set

I'm trying to show that $\mathbf{W} = \mathrm{span} \{ 1+x, 1-x^2 \}$ where $\mathbf{W} = \{a + b + ax - bx^2 \mid a,b \in\mathbb{R}\}$.

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what's your definition of "spanned vector space" ? –  mercio Mar 23 '11 at 18:05

## 1 Answer

Write $a + b + ax - bx^2 = u(1+x)+v(1-x^2)$. Solve for $u$ and $v$, given $a$ and $b$. This proves one inclusion. Do the reverse to prove the other inclusion.

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Doing so proves only half the equality, no? –  Larry Denenberg Mar 23 '11 at 19:39
@Larry, yes. I've edited my answer. Thanks for the nudge. –  lhf Mar 23 '11 at 19:57