Trying to learn linear algebra on my own

[InbredDummy]

Waddup everyone

I'm trying to learn linear algebra on my own. I'm a freshman, blah blah, who really cares, now to the math :)

1) Prove that if (v1,v2,v3) span V, then (v1 + 2v2, v2 - v3, v3) also span V

2)Prove that if (v1,v2,v3) is linearly independent, then (v1 + 2v2, v2 - v3, v3) is also linearly independent

3) let P(|F) be the space of polynomials with coefficients in |F
(i) Find two different subspaces of P(|F)
(ii) Find an infinite dimensional proper subspace of P(|F)

4) Let U,V be subspaces of F^7, such that U and the direct sum of V = F^7. if dim U=3, show that dim V=4.
(ii) does this remain true if all we know is that U+V=F^7 ?

i'm new at this, I've been reading the textbooks and what have you, but I'm not seeing the answers.

thx

 

[buddyholly9999]

From what I've seen on most of the homework posts...people usually want you to show what work you've done already..

 

[InbredDummy]

im really stuck. i posted to see how more seasoned students would handle the problems and hopefully i can build from there.

 

[matt grime]

You see, one of the things you need to realize is that 'seeing the answer' isn't how you get the answer. You have to write things down, puzzle over them for a while, check you understand the definitions and so on. Write out the definition of 'span' and try to figure out how to do 1) for yourself. If you know the definition of span and are happy with it then the answer will suddenly drop out.

For instance, the proof of 2) is gotten simply by writing out the definition of linearly dependent (i.e. the negation of linearly independent).

 

[HallsofIvy]

Waddup everyone

I'm trying to learn linear algebra on my own. I'm a freshman, blah blah, who really cares, now to the math :)

1) Prove that if (v1,v2,v3) span V, then (v1 + 2v2, v2 - v3, v3) also span V

What is the definition of "span V"?

2)Prove that if (v1,v2,v3) is linearly independent, then (v1 + 2v2, v2 - v3, v3) is also linearly independent

What is the definition of "linearly independent"?

3) let P(|F) be the space of polynomials with coefficients in |F
(i) Find two different subspaces of P(|F)
(ii) Find an infinite dimensional proper subspace of P(|F)

What are some members of P(|F)? (In other words how is it defined? What is the definition of "subspace"?

Do you see a pattern here?

4) Let U,V be subspaces of F^7, such that U and the direct sum of V = F^7. if dim U=3, show that dim V=4.
(ii) does this remain true if all we know is that U+V=F^7 ?

i'm new at this, I've been reading the textbooks and what have you, but I'm not seeing the answers.

thx[/QUOTE]

 

[InbredDummy]

3) let P(|F) be the space of polynomials with coefficients in |F
(i) Find two different subspaces of P(|F)
(ii) Find an infinite dimensional proper subspace of P(|F)

k, here is what i got so far:
(i) let U=a+b+cx^2
and let V=cx^2

they are both 2 dimensional subspaces of P(|F) but are obviously disjoint and therefore different

(ii)
no clue here on this one. how can it be infinitely dimensional but able to be written down? would it just be x^(infinity)?

 

[matt grime]

U is a polynomial, as is V. If you mean a set of polynomials say so. If V is supposed to be the set of polys { cx^2 : c in F} then that is not a two dimensional vector space. What is the definition of dimension? It is not, as you seem to think, 'the exponent of x in a polynomial'. You see why we hammered home the message that you need to know what the definitions are?

HINT: the even integers are an infinite subset of all integers.