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Determine whether the following polynomials span $\mathbb P_2$

$p_1 = 1+x+2x^2$

$p_2 = 3+x$

$p_3 = 5-x+4x^2$

$p_4 = -2- 2x +2x^2$

Attempt:

I created the following augmented matrix: $$ \begin{bmatrix} 1 & 3 & 5 & -2 & a \\ x & x & -x & -2x&bx\\ 2x^2 & 0 & 4x^2 & 2x^2 & cx^2\\ \end{bmatrix} $$ which reduces to:

$$ \begin{bmatrix} 1 & 3 & 5 & -2 & a \\ 0 & 1 & 3 & 0 &(a-b)/2\\ 0 & 0 & 1 & 1/2 & (c+a-3b)/2\\ \end{bmatrix} $$

Now, I believe that the system is consistent and has infinitely many solutions so the polynomials span $\mathbb{P_2}$.

Is this approach correct?

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  • $\begingroup$ I'm not sure that " I believe that the system is consistent and has infinitely many solutions" is persuasive. In this case, it's easy enough to just work with linear combinations of the vectors themselves. Subtracting the fourth from the first we see that $3+3x$ is in the span. Combine that with the second to deduce that $1$ and $x$ are in the span. Conclude from there. $\endgroup$ – lulu Jan 24 '20 at 13:02
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    $\begingroup$ You argument is correct. You even see that $p_1, p_2, p_3$ form a basis of $\mathbb P_2$. Infinitely many solutions are obtained because of the existence of a fourth polynomial $p_4$. $\endgroup$ – Paul Frost Jan 24 '20 at 13:17
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If $P_2$ is the vector space of real polynomials of order smaller or equal to 2, then you can think of the given polynomials as vectors in $\mathbb {R^3}$ as; $p_1=(1,1,2)$, $p_2=(3,1,0)$, $p_3=(5,-1,4)$, $p_4=(-2,-2,2)$ (Why?). Then check the rank of the following matrix: \begin{pmatrix} 1 & 1 & 2\\ 3 & 1 & 0\\ 5 & -1 & 4\\ -2 & -2 & 2 \end{pmatrix}

If the rank is equal to 3, then the answer is affirmative.

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  • $\begingroup$ why is my method not correct? The rank of my matrix is also 3. $\endgroup$ – Archer Jan 24 '20 at 15:02
  • $\begingroup$ Why'd you choose a 4*3 matrix instead of a 3*4 matrix? Also, why should the rank be three? $\endgroup$ – Archer Jan 25 '20 at 11:24

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