Question

Linear Algebra: Spanning Sets

Answer 1

Which of the sets that follow are spanning sets for P(3)? Justify your answers.
Okay, I'm pretty awful at this so...first one
\[(1, x^{2}, x^2-2)\] Which in order for it to be shown to span all polynomials of Degree 3 it would have to \[Span[c_1(1)+c_2(x^2)+c_3(x^2-2)] = a_0+a_1x+a_2x^2+a_3x^3\] right?
Now my problem then would be how I would begin to prove/justify that. I'm hoping if I can get some help on this one I can figure the rest out on my own...

Answer 2

So you have the polinomies,
\[p_1=1\\
p_2=x^2\\
p_3=x^3-2\]These are like usual vectors, so you must prove that they are linearly independent. Start with the general form y=ax^3+bx^2+cx+d
\[\left[\begin{matrix} 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0\\1 & 0 & 0 & -2 \end{matrix}\right]\]
And find the rank of this matrix. If the rank is 3, then it spans P_3.

Answer 3

Okay. so, since there isn't a rank 3 term in the original, then this set cannot span P(3)? (rank and degree mean the same thing concerning polynomials right?)

Answer 4

Exact.

Answer 5

I have a mistake in my phrase. The matrix should have rank 4 as you said in order to span P3.

Answer 6

It can only span \[
\begin{bmatrix}
x^0\\
x^2
\end{bmatrix}
\]

Answer 7

Okay, then what about the second set in the problem: \[(2, x^2, x, 2x+3)\] Because that doesn't have any terms higher than rank 2, but my book says it spans P(3)?

Answer 8

It cannot span P3, for sure.

Answer 9

What does \(P3\) mean?

Answer 10

Hmm, my book says that and the next set span P(3). and none of the given sets are higher than rank 3.
Wio, P(3) is all polynomials of degree 3

Answer 11

(I'm pretty sure.)

Answer 12

You're not writing the problem down correctly, period.

Answer 13

Well, if P3 represents polinomy of order 3, then the set you says cannot span P3, as it has not any x^3.

Answer 14

Um...How should I be writing it down then? Cus I just copied the question right out of the book...

Answer 15

@blarghhonk8 do you see that the set cannot span P3 (ax^3+bx^2+cx+d) because there is no x^3?

Answer 16

Yeah, that makes sense to me. But my book gives four sets: \[1. (1,x^2,x^2-2)\]\[2.(2, x^2,x,2x+3)\]\[3.(x+2,x+1, x^2-1)\]\[4.(x+2,x^2-1)\]And says 1 and 4 do not span p(3), but 2 and 3 do span P(3).
So, since none of them have any term of degree 3 it seems there's something we're missing :/

Answer 17

Then I think wio is right.

Answer 18

P3 does not means polinomies of degree 3, but this other thing Pn are polinomies of degree n-1.

Answer 19

Then my book has a major major typo in it cus the question word for word is:
Which of the sets that follow are spanning sets for P(3)? Justify your answer. and then goes on to list the 4 above.
Okay. that would make sense then.

Answer 20

Yes all depends on how your book defines P(3), but as we see, is P(n) means degree n-1.
;)