# Linear algebra, subspace of a vector space?

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!

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#### jbunniii

Homework Helper
Gold Member
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

Homework Helper
Gold Member
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?

#### Mark44

Mentor
What exactly were you pouring over your book? Coffee? Beer?? And why?
Yeah, not a good idea - the pages will stick together. :tongue:

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