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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}
I'm pretty sure you mean, S = {1 + x^{2}, x + x^{3}}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}
Prob. 15 is almost identical to prob. 13. The polynomials in P_{3} are essentially the same as vectors in R^{4}. For example, 1 + 2x^{2} <---> <1, 0, 2, 0>.I scanned it in. Its problem # 15
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.
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?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.
But the second part. Augment S to form a basis S' for P_{3} , that would still be in the form of a polynomial?Yes.
Ok, to verify 13 is done properly?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 P_{3}.