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Linear algebra, subspace of a vector space?

  • Thread starter toyotadude
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1. The problem statement, all variables and given/known data

1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?

2) The set H of all polynomials p(x) = a+bx^3, with a,b in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?

2. Relevant equations

Eh.. not sure?

3. The attempt at a solution

Once more, not too sure. I've been pouring over my Linear Algebra book, but it seems so abstract... I was under the impression that as long as the polynomial didn't have a lower power than the vector space [number] that the polynomial would be in the subspace of the given vector space :\

Does the coefficient have anything to do with it?

Some properties (if they help?): A subspace of a vector space V is a subset H of V that has 3 properties:
a) The zero vector if V is in H.
b) H closed under vector addition
3) H closed under scalar multiplication..

I've already gotten #1 wrong (the answer was false) - I'd like to know why though :(

Any help would be awesome!
 

jbunniii

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1. The problem statement, all variables and given/known data

1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?
Some properties (if they help?): A subspace of a vector space V is a subset H of V that has 3 properties:
a) The zero vector if V is in H.
b) H closed under vector addition
3) H closed under scalar multiplication..

I've already gotten #1 wrong (the answer was false) - I'd like to know why though :(
Consider property (a). Can the zero vector (polynomial in this case) be written in the form ##p(x) = a + x^3##?

Also consider property (b). If I have two polynomials of the form ##a + x^3##, and add them together, is the result also of the form ##a + x^3##?

Finally, consider property (c). If I multiply a polynomial of the form ##a + x^3## by an arbitrary scalar, say ##2##, is the result of the form ##a + x^3##?
 

LCKurtz

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Once more, not too sure. I've been pouring over my Linear Algebra book, but it seems so abstract.
What exactly were you pouring over your book? Coffee? Beer?? And why? :confused:
 

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