Is U+W=V?

[TheWrathOfMath]

V=P2[x]
U={p(x)∈V | a+b+c=0}
W={p(x)∈V |p(0)=p(1)}

a) Is U+W=V ?
b) Is U⊕W=V ?

Is what I did so far correct for (a), and if so, how do I proceed?

U={p(x)∈V | a+b+c=0}
=>
U={a+bx+cx^2 | a+b+c=0}
=>
U={a+bx+(-a-b)x^2 | a,b ∈ F}


W={p(x)∈V |p(0)=p(1)}
=>
W={a+bx+cx^2 | a+b(0)+c(0)^2 = a+b(1)+c(1)^2}
=>
W={a+bx+cx^2 | a=a+b+c}
=>
W={a+bx+cx^2 | c= -b}
=>
W={a+bx+(-b)x^2 | a,b ∈ F}


U+W = a+bx+(-a-b)x^2 + a+bx+(-b)x^2
= 2a+2bx+(-a-2b)x^2
.
.
.

How do I proceed from here?

I know that P2(x) is of the form: a+bx+cx^2.

 

[blamocur]

In your expression for [imath]U+W[/imath] why do you use the same a's, b's and c's for both polynomials (one from U, another from W)?

Personally, I would not drag x's and x^2 around, just use equations for a,b,c and see what kind of subspaces they define.
E.g. [imath]U=\{(u,v,-u-v)\}, \;W=\{(a,b,-b)\}[/imath], then [imath]U+W[/imath] consists of all vectors which can be represented by sums of vectors from [imath]U[/imath] and [imath]W[/imath].

An alternate approach is to look at polynomials themselves: if [imath]a+b+c=0[/imath] what can you tell about [imath]p(x) = a+bx+cx^2[/imath]? (Hint: what do we know about [imath]p(1)[/imath]?)

 

[TheWrathOfMath]

In your expression for [imath]U+W[/imath] why do you use the same a's, b's and c's for both polynomials (one from U, another from W)?

Personally, I would not drag x's and x^2 around, just use equations for a,b,c and see what kind of subspaces they define.
E.g. [imath]U=\{(u,v,-u-v)\}, \;W=\{(a,b,-b)\}[/imath], then [imath]U+W[/imath] consists of all vectors which can be represented by sums of vectors from [imath]U[/imath] and [imath]W[/imath].

An alternate approach is to look at polynomials themselves: if [imath]a+b+c=0[/imath] what can you tell about [imath]p(x) = a+bx+cx^2[/imath]? (Hint: what do we know about [imath]p(1)[/imath]?)

So I fixed it and found out that the expressions for any vectors belonging to U and W respectively are as followed:

w∈W :
w= (alpha)+(beta)x+(-beta)x^2

u∈U:
u = a+bx+(-a-b)x^2

Can I say that the basis for U= {1-x^2, x-x^2} and the basis for W = {1, x-x^2}?

And then in order to find the basis for U+V, I will put the basis for U and the basis for W as rows in a matrix and transform it to REF?

 

[blamocur]

Can I say that the basis for U= {1-x^2, x-x^2} and the basis for W = {1, x-x^2}?

Looks correct to me.

And then in order to find the basis for U+V, I will put the basis for U and the basis for W as rows in a matrix and transform it to REF?

I don't know what REF means, but you have 4 vectors -- how many of them are linearly independent?

 

[TheWrathOfMath]

I don't know what REF means,

REF = Reduced Echelon Form.
The intention was to use this method to omit vectors which are linear combinations of the others and be left with only linearly independent vectors. It is unnecessary in this case, though, since it is very easy to find the linearly independent vectors.
I apologize for not being clear.

but you have 4 vectors -- how many of them are linearly independent?

I can obviously omit one of the x-x^2, so I am left with three linearly independent vectors which constitute a basis for U+V.

 

[blamocur]

REF = Reduced Echelon Form.
The intention was to use this method to omit vectors which are linear combinations of the others and be left with only linearly independent vectors. It is unnecessary in this case, though, since it is very easy to find the linearly independent vectors.
I apologize for not being clear.



I can obviously omit one of the x-x^2, so I am left with three linearly independent vectors which constitute a basis for U+V.

Can you show that they are linearly independent?

 

[TheWrathOfMath]

Can you show that they are linearly independent?

Yes, by using a matrix and showing that:

c1v1+c2v2+c3v3=0, and c1=c2=c3=0.

I will get that the basis for U+W= {1, 1-x^2, x-x^2}.

The basis for V is {1, x, x^2}.

The original question was "is U+W=V"?

How do I proceed from here?
Can I claim that {1, 1-x^2, x-x^2} =/= {1, x, x^2} or not?

 

[blamocur]

Yes, by using a matrix and showing that:

c1v1+c2v2+c3v3=0, and c1=c2=c3=0.

I will get that the basis for U+W= {1, 1-x^2, x-x^2}.

The basis for V is {1, x, x^2}.

The original question was "is U+W=V"?

How do I proceed from here?
Can I claim that {1, 1-x^2, x-x^2} =/= {1, x, x^2} or not?

They are different bases but you can show that they are equivalent (i.e., their spans are the same) if you show how to express the vectors in the second basis (for [imath]V[/imath]) as linear combinations of vectors from the first one (for [imath]U+W[/imath]) -- wouldn't you agree?

Or you could show that your basis for [imath]U+W[/imath] is