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

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In summary, to determine if the polynomial $3x^2+2x-1$ is in the span $\{x^2+x-1,x^2-x+2,1\}$, one needs to compare the coefficients of $x^2$, $x$, and the constant terms. This will give a system of equations in $c_1$, $c_2$, and $c_3$. If the system is consistent, then the given polynomial is in the span, otherwise not. The example provided shows the process of solving the system of equations to determine if a polynomial is in a given span.
  • #1
karush
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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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  • #2
karush said:
$c_1(x^2+x-1)+c_2(x^2-x+2,1)=3x^2+2x-1$

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.
 
  • #3
https://www.physicsforums.com/attachments/9043

ok here is an example but I don't see how they got the numbers"
 
  • #4
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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  • #5
$c_1(x^2+x-1)+c_2(x^2-x+2)+c_3\ \equiv\ 3x^2+2x-1$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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Related to 11.1 Determine if the polynominal.... is the span of (....,....)

1. What does it mean for a polynomial to be the span of two other polynomials?

For a polynomial to be the span of two other polynomials, it means that it can be written as a linear combination of those two polynomials. This means that the polynomial can be expressed as a sum of multiples of the two polynomials, with different coefficients.

2. How do you determine if a polynomial is the span of two other polynomials?

To determine if a polynomial is the span of two other polynomials, you can use the method of substitution. Substitute the values of the two polynomials into the given polynomial and see if it results in the given polynomial. If it does, then the polynomial is the span of the two polynomials.

3. Can a polynomial be the span of more than two polynomials?

Yes, a polynomial can be the span of more than two polynomials. In general, a polynomial can be the span of any number of polynomials, as long as it can be expressed as a linear combination of those polynomials.

4. What is the significance of determining if a polynomial is the span of two other polynomials?

Determining if a polynomial is the span of two other polynomials can help in understanding the relationship between different polynomials. It can also be useful in solving systems of equations and finding solutions to problems in linear algebra and other areas of mathematics.

5. What are some real-world applications of determining if a polynomial is the span of two other polynomials?

Some real-world applications of determining if a polynomial is the span of two other polynomials include predicting the behavior of systems in physics and engineering, analyzing data in statistics, and solving optimization problems in economics and finance.

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