Linear algebra, subspace of a vector space?

[toyotadude]

Homework Statement

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?

Homework Equations

Eh.. not sure?

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]

Homework Statement

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]

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]

What exactly were you pouring over your book? Coffee? Beer?? And why?

Yeah, not a good idea - the pages will stick together.