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[SOLVED] 11.1 Determine if the polynominal.... is the span of (...,...)

karush

Well-known member
Jan 31, 2012
2,657
Determine if the polynomial
$3x^2+2x-1$
is the $\textbf{span}\{x^2+x-1,x^2-x+2,1\}$


ok from examples it looks like we see if there are scalars such that
$c_1(x^2+x-1)+c_2(x^2-x+2,1)=3x^2+2x-1$


so far not sure how this is turned into a simultaneous eq

I did notice that it is common to get over 100 views on these DE problems
so thot it would be good to show sufficient steps
 
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Olinguito

Well-known member
Apr 22, 2018
251
$c_1(x^2+x-1)+c_2(x^2-x+2,1)=3x^2+2x-1$
It should be
$$c_1(x^2+x-1)+c_2(x^2-x+2)+c_3\ \equiv\ 3x^2+2x-1$$
Expand out the LHS and compare coefficients. This will give you a system of equations in $c_1,c_2,c_3$. If the system is consistent (i.e. has at least one solution) then the given polynomial is in the given span, otherwise not.
 

karush

Well-known member
Jan 31, 2012
2,657

Olinguito

Well-known member
Apr 22, 2018
251
That solution is for the problem of determining if this polynomial
$$2x^2+x+1$$
is in $\mathrm{Span}\{x^2+x,x^2-1,x+1\}$.

In your OP, you have this:
$$c_1(x^2+x-1)+c_2(x^2-x+2)+c_3\ \equiv\ 3x^2+2x-1$$
so comparing coefficients of $x^2$, $x$ and the constant terms should give you
$$\begin{array}{rcrcrcr}c_1 & +&c_2 && &=& 3 \\ c_1 &-&c_2 && &=& 2 \\ c_1 &+&2c_2 &+&c_3 &=& -1\end{array}.$$
 
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karush

Well-known member
Jan 31, 2012
2,657
$c_1(x^2+x-1)+c_2(x^2-x+2)+c_3\ \equiv\ 3x^2+2x-1$


ok I got ....


\begin{array}{rcrcrcr}c_1 & +&c_2 && &=& 3 \\ c_1 &-&c_2 && &=& 2 \\ -c_1 &+&2c_2 &+&c_3 &=& -1\end{array}

could we multiply thru the $R_3$ with -1

well anyway got this ...

$\left[ \begin{array}{ccc|c} 1 & 0 & 0 & \frac{5}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & - \frac{9}{2} \end{array} \right]$
 
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