- #1

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Let S be a subspace of P

_{3 consisting of all polynomials of the form ax2+bx+2a+3b. Find a basis for S. I am not sure where to start.}

- Thread starter Dustinsfl
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- #1

- 699

- 5

Let S be a subspace of P

- #2

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Find a generating set for this subspace. How about {x

- #3

- 41

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Try "factoring out a". It can't factor out of everything, but try factoring it out of all the terms you can.

--------

If that doesn't make sense, try to get a feel for what the subspace is by choosing values for a and b.

For instance, which element does a = 2, b = 0 give you?

--------

If that doesn't make sense, try to get a feel for what the subspace is by choosing values for a and b.

For instance, which element does a = 2, b = 0 give you?

Last edited:

- #4

- 699

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Then set that equal to 0? Also factor out b the same way?

- #5

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However, yes, if you correctly factor out the "a"s and the "b"s, you should start to see the answer.

- #6

- 699

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a(x^{2}+2)+b(x+3)

- #7

- 699

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I can then set up the Wronskian.

- #8

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Do you know what to do now, or do you still have questions?

- #9

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I am still confused but I evaluated the Wronskian. How can I come up with the basis now?

W=x^2+6x-2

W=x^2+6x-2

- #10

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(1)

Every element of S can be written as

a(x

This shows that (x

(2)

The Wronskian of these two elements is x

A linearly independent generating set is precisely a basis, so these two polynomials form your basis.

- #11

- 699

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That is it?

Thanks.

Thanks.

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