- #1
RJLiberator
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Homework Statement
Determine if the following sets are bases for [itex]P_2(R)[/itex]
b) [itex](1+2x+x^2, 3+x^2,x+x^2)[/itex]
d) [itex](-1+2x+4x^2, 3-4x-10x^2,-2-5x-6x^2)[/itex]
Homework Equations
Bases IF Linear Independence AND span(Set)=[itex]P_2(R)[/itex]
RREF = Reduced Row Echelon Form
The Attempt at a Solution
My first question here regards an understanding of notation.
So for each B and D I did worked it out to be in Row Echelon Form and found linear independce. Permitting I did the calculations correctly, is it safe to say that these are basis for [itex]P_2(R)[/itex] as the amount of terms in the set is 3 and since 3-1 = 2 P_2 is safe and these sets span P_2(R)?
Second question: For b) I was able to get the RREF rather easily from the matrix:
\begin{pmatrix}
1 & 2 & 1 & 0\\
1 & 0 & 3 & 0\\
1 & 1 & 0 & 0
\end{pmatrix}
This should be linear independent in RREF, and so it is a bases. An answer key says "no" this is not a bases. However, the answer key can be wrong. Is there anything I did wrong here in setting this problem up for b) or perhaps my understanding of span?Third question: For d) I was checking a solution and the solution had set up the matrix differently then what I had expected. They set up the matrix as such:
\begin{pmatrix}
-1 & 3 & -2 & 0\\
2 & -4 & -5 & 0\\
4 & -10 & -6 & 0
\end{pmatrix}
As you can see, since d) [itex](-1+2x+4x^2, 3-4x-10x^2,-2-5x-6x^2)[/itex], they switched the columns and rows that I'm traditionally used to. Is this ok to do? The normal way I do it, like my example in b, is difficult to get into RREF, however this way is rather easy.
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