[SOLVED] Prove polynomials are a generating system (vector spaces)

**bmp05** · May 3rd 2009, 11:50 AM

Let V be the vector space of the polynomials of degree $\leq$ 2 over $\mathbb{R}$ .

Prove that the polynomials:

$p_1 = 2T^2 + T + 1$
$p_2 = 4T^2 + T$
$p_3 = -2T^2 + 2T + 1$

span, or are a generating system of, V.

If I solve the system of linear equations- how do I prove that they span V?
Thanks.

**ThePerfectHacker** · May 3rd 2009, 12:19 PM

Originally Posted by bmp05

Let V be the vector space of the polynomials of degree $\leq$ 2 over $\mathbb{R}$ .

Prove that the polynomials:

$p_1 = 2T^2 + T + 1$
$p_2 = 4T^2 + T$
$p_3 = -2T^2 + 2T + 1$
.

You just need to show these polynomial are linearly independent.
Say that $a_1p_1 + a_2p_2+a_3p_3 = \bold{0}$ where $\bold{0}$ is the zero-polynomial.
This means that,
$2a_1+4a_2-2a_3=0$
$a_1 + \ \ \ a_2 + 2a_3 = 0$
$a_1 + \ \ \ \ \ \ \ \ \ + a_3 = 0$

Argue that $a_1=a_2=a_3=0$ is the only solution.

**Twig** · May 3rd 2009, 12:24 PM

Write the polynomials as vectors in $\mathbb{R}^{3}$ .

$A= \left[\begin{matrix} 1 & 0 & 1\\ 1 & 1 & 2 \\ 2 & 4 & -2 \end{matrix}\right]$

Row operations will show that this matrix A has 3 pivot positions. So it therefore spans $\mathbb{R}^{3}$ and by the isomorphism between $\mathbb{R}^{3} \mbox{ and } \mathbb{P}_{3}$ also $\mathbb{P}_{3}$ .

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