- #1

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## Homework Statement

Given S = (1+x

^{2}, x +x

_{3}

And augment S to form a Basis S' of P

_{3}

## The Attempt at a Solution

0 + 0x + 0x

^{2}+ 0x

^{3}= a(1+x

^{2})+b(x +x

^{3})

= a + ax

^{2}+ bx + bx

^{3}

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- Thread starter judahs_lion
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- #1

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Given S = (1+x

And augment S to form a Basis S' of P

0 + 0x + 0x

= a + ax

- #2

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Isn't S dependent? X + X^{3} = (1 + X^{2})X

- #3

Mark44

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I'm pretty sure you mean, S = {1 + x## Homework Statement

Given S = (1+x^{2}, x +x_{3}

It would be helpful for you to state the complete problem. My guess is that it is two parts:And augment S to form a Basis S' of P_{3}

## The Attempt at a Solution

0 + 0x + 0x^{2}+ 0x^{3}= a(1+x^{2})+b(x +x^{3})

= a + ax^{2}+ bx + bx^{3}

a) Prove that the functions in S = {1 + x

b) Augment S to a set S' that is a basis for P

For a, how is linear independence defined? From your work above, I'm not sure that you know. The definitions for linear independence and linear independence are similar, and there is a subtlety that students often don't grasp.

For b, have you learned about the Gram-Schmidt process?

- #4

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- #5

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Haven't gotten to Gram-Schmidt process

- #6

Mark44

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Prob. 15 is almost identical to prob. 13. The polynomials in PI scanned it in. Its problem # 15

- #7

Mark44

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This isn't the definition, and besides, a definition of a term ought not use the same term in the definition. Look in your book and see how it defines linear independence.Linear independence mean the members of a set of vectors are independent of each other.

This is a necessary condition for linear independence, but it is not sufficient. For example, consider the set {<1, 0, 0>, <0, 1, 0>, <1, 1, 0>}. None of these vectors is a multiple of any other vector in the set, yet these vectors are not linearly independent.None is a multiple of the other.

- #8

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This isn't the definition, and besides, a definition of a term ought not use the same term in the definition. Look in your book and see how it defines linear independence.

This is a necessary condition for linear independence, but it is not sufficient. For example, consider the set {<1, 0, 0>, <0, 1, 0>, <1, 1, 0>}. None of these vectors is a multiple of any other vector in the set, yet these vectors are not linearly independent.

Thanks for pointing out that so , what i need to do is a transformtion as i did in the attached. Then the rest is just as problem 13?

- #9

Mark44

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Yes.

- #10

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Yes.

But the second part. Augment S to form a basis S' for P

- #11

Mark44

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- #12

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_{3}.

Ok, to verify 13 is done properly?

- #13

Mark44

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- #14

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S' = {1+x^{2}, x+x^{3}, 1, x }

- #15

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- #16

Mark44

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