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Consider the vector space of real-valued polynomials of the power not larger than 3:

$P_3(x) = a_0 + a_1x + a_2x^2 + a_3x^3$:

(a) Write down a set of functions that form a basis of this vector space.

I have no idea how to do this. Can someone please help?

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5  
Hint: $P_3(x) = a_0\cdot1 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot x^3$. –  njguliyev Oct 19 '13 at 21:05

1 Answer 1

Hint: you need every function of the form

\begin{align*} P(x) &= a_0 + a_1x + a_2x^2 + a_3x^3 \\ &= a_0x^0 + a_1x^1 + a_2x^2 + a_3x^3. \end{align*}

written in the form

$$P(x) = a_0 P_0(x) + a_1 P_1(x) + a_2 P_2(x) + a_3 P_3(x),$$

where $P_k(x)$, $k=0,\dots,3$, are polynomials that you're looking for, and $a_k \in \mathbb{R}$, $k=0,\dots,3$, are given scalars that define $P(x)$.

Further hint: you can choose each $P_k$ to be a polynomial of degree $k$.

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So would it be $(1, x, x^2, x^3)$ with a dimension of 4? –  guest89 Oct 19 '13 at 22:37
    
Yes. If you're not sure, notice that a) they span what you want, and b) that set has the dimension of the space. Hence, it is a basis. –  Vedran Šego Oct 19 '13 at 23:04

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