Prove Independence of P3 Basis S

[judahs_lion]

Homework Statement

Given S = (1+x², x +x₃

And augment S to form a Basis S' of P₃

The Attempt at a Solution

0 + 0x + 0x² + 0x³ = a(1+x²)+b(x +x³)

= a + ax² + bx + bx³

 

[judahs_lion]

Isn't S dependent? X + X³ = (1 + X²)X

 

[Mark44]

Homework Statement

Given S = (1+x², x +x₃

I'm pretty sure you mean, S = {1 + x², x + x³}

And augment S to form a Basis S' of P₃

The Attempt at a Solution

0 + 0x + 0x² + 0x³ = a(1+x²)+b(x +x³)

= a + ax² + bx + bx³

It would be helpful for you to state the complete problem. My guess is that it is two parts:
a) Prove that the functions in S = {1 + x², x + x³} are linearly independent.
b) Augment S to a set S' that is a basis for P₃.

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?

 

[judahs_lion]

I scanned it in. Its problem # 15

 

[judahs_lion]

Linear independence mean the members of a set of vectors are independent of each other. None is a multiple of the other.
Haven't gotten to Gram-Schmidt process

 

[Mark44]

I scanned it in. Its problem # 15

Prob. 15 is almost identical to prob. 13. The polynomials in P₃ are essentially the same as vectors in R⁴. For example, 1 + 2x² <---> <1, 0, 2, 0>.

 

[Mark44]

Linear independence mean the members of a set of vectors are independent of each other.

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.

None is a multiple of the 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.

 

[judahs_lion]

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?

 

[Mark44]

Yes.

 

[judahs_lion]

Yes.

But the second part. Augment S to form a basis S' for P₃ , that would still be in the form of a polynomial?

 

[Mark44]

You can use the augmented basis you found in #13, and "untransform" the vectors to get the other two polynomials you need for a basis for P₃.

 

[judahs_lion]

You can use the augmented basis you found in #13, and "untransform" the vectors to get the other two polynomials you need for a basis for P₃.

Ok, to verify 13 is done properly?

 

[Mark44]

Sure, those vectors are linearly independent, one of many possible sets of four vectors that span R^4.

 

[judahs_lion]

S' = {1+x², x+x³, 1, x }

 

[judahs_lion]

Would Reducing the Matrix (1, 0; 0,1; 1, 0; 0,1) to (1,0; 0,1; 0,0; 0,0) have been another way to prove independence?

 

[Mark44]

Yes, but with just two vectors, that's overkill. Two vectors are linearly independent as long as neither one is a multiple of the other. If you have three vectors, though, it's not as obvious. I gave you an example of this in another thread.