U∩W

[TheWrathOfMath]

V= P4[x]

The the basis for W, which is the linear subspace of all polynomials in V such as that when multiplied by (x-2), the resulting polynomial will include only odd degree exponents. is {2x+x^2, 2x^3+x^4}

The basis for U={q(x)∈V | q(-1)=0} ⊆ V=P4[x] is {1-x^4, x+x^4, x^2-x^4, x^3+x^4} .
I found it by letting q(x) = f+gx+hx^2+jx^3+kx^4, used the constraint q(-1)=0 to find that k= -f+g-h+j, and substituted it in q(x), simplified, and obtained the basis.

I was asked to find the intersection of U and W.

I let v∈ U∩W, therefore:

v∈W, hence:
v= c(2x+x^2)+e(2x^3+x^4)

v∈U, hence:
a(1-x^4)+b(x+x^4)+c(x^2-x^4)+d(x^3+x^4)
*Notice that I changed the coefficients from f,g,h,j to a,b,c,d.

v=v, therefore v-v=0 :

a(1-x^4)+b(x+x^4)+c(x^2-x^4)+d(x^3+x^4)-c(2x+x^2)-e(2x^3+x^4) = (0,0).

Now I obtained the following equations:

a+bx+cx^2+dx^3-2cx-2ex^3=0
-ax^4+bx^4-cx^4+dx^4-cx^2-ex^4=0

I thought about attempting to solve the system of equations and expressing some variables using other (free) variables.

How do I do that using a matrix?
I am slightly confused about how will the matrix look like.
Ax=b
Will A be a matrix of the coefficients, x a 1x2 matrix of the variables, and b a 1x2 zero matrix?
If so, don't I need to rearrange my equations, since I have the coefficient c, for instance, appearing in two different terms in the first and second equations?

For instance:

a+bx+c(x^2-2x)+dx^3+e(-2x^3) = 0
a(-x^4)+b(x^4)+c(-x^4-x^2)+d(x^4)+e(-x^4) =0

I am pretty sure it is wrong, though, because then the matrix A will simply contain two rows of only the entry "1"...

Perhaps I need to group and take out like variables, so that:

1(a)+x(b-2c)+x^2(c)+x^3(d-2e) = 0
x^2(-c)+x^4(-a+b-c+d-e) = 0

 

[blamocur]

I've noticed that you are using the same variable `c` in expressions for `v` in U and W.

 

[TheWrathOfMath]

I've noticed that you are using the same variable `c` in expressions for `v` in U and W.

Right. I wrote: *Notice that I changed the coefficients from f,g,h,j to a,b,c,d.

 

[blamocur]

Right. I wrote: *Notice that I changed the coefficients from f,g,h,j to a,b,c,d.

I am not sure how this is related to my note. Just to make sure we are on the same page, I've marked duplicated variable in bold red in the quote below:

v∈W, hence:
v= c(2x+x^2)+e(2x^3+x^4)

v∈U, hence:
a(1-x^4)+b(x+x^4)+c(x^2-x^4)+d(x^3+x^4)

 

[TheWrathOfMath]

I am not sure how this is related to my note. Just to make sure we are on the same page, I've marked duplicated variable in bold red in the quote below:

Perhaps I should have continued using the variables f,g,h,j, then.
I thought that I could use the same variables for both equations since v is a vector in both U and W, but perhaps I can't do that.

 

[blamocur]

Perhaps I should have continued using the variables f,g,h,j, then.
I thought that I could use the same variables for both equations since v is a vector in both U and W, but perhaps I can't do that.

I think it is important to start with different sets of variables unless you can justify reusing the same variable in both expressions. I did not notice any justifications there.

 

[TheWrathOfMath]

I think it is important to start with different sets of variables unless you can justify reusing the same variable in both expressions. I did not notice any justifications there.

I suppose that a better question is how to find the basis of the intersection of U and W, given that the basis for U is
{1-x^4, x+x^4, x^2-x^4, x^3+x^4}, and the basis for W is {2x+x^2, 2x^3+x^4}.

If I use a set of different variables, do I just do what I thought about doing? Subtract the general expression of W from the general expression of U, equate the equation to zero, create two equations and solve them using a matrix?

 

[blamocur]

I suppose that a better question is how to find the basis of the intersection of U and W, given that the basis for U is
{1-x^4, x+x^4, x^2-x^4, x^3+x^4}, and the basis for W is {2x+x^2, 2x^3+x^4}.

If I use a set of different variables, do I just do what I thought about doing? Subtract the general expression of W from the general expression of U, equate the equation to zero, create two equations and solve them using a matrix?

I don't see any problems with this approach, but why do you expect only two equations. Also, the equations I got were simple enough not to require using matrices.

 

[TheWrathOfMath]

I don't see any problems with this approach, but why do you expect only two equations. Also, the equations I got were simple enough not to require using matrices.

This is what I got:

Let v∈ U∩W, therefore:

v∈W, hence:
v= c(2x+x^2)+e(2x^3+x^4)

v∈U, hence:
f(1-x^4)+g(x+x^4)+h(x^2-x^4)+j(x^3+x^4)

I distributed, subtracted the two expression and equated the equation to zero:

0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3-(-f+g-h+j)x^4

What do I do from here?
Do I rearrange it such that like-terms in terms of variables are grouped together, as in x(something)+x^2(something else)+ . . .
or did I NOT have to distribute the two expressions in the first place and just end up with an equation in which the coefficients are taken out?
For instance, c(something)+e(something else)+ . . .

 

[blamocur]

I don't know if this is the only good approach, but when polynomial is 0 this means that very coefficient is 0. Now you get 5 equations for a bunch of variables, but this equations are actually quite simple once you take a closer look.

 

[TheWrathOfMath]

I don't know if this is the only good approach, but when polynomial is 0 this means that very coefficient is 0. Now you get 5 equations for a bunch of variables, but this equations are actually quite simple once you take a closer look.

I see.
And what about the other question I asked specifically about the equations?

Do I rearrange 0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3-(-f+g-h+j)x^4
such that like-terms in terms of variables are grouped together, as in x(something)+x^2(something else)+ . . .
or did I NOT have to distribute the two expressions in the first place and just end up with an equation in which the coefficients are taken out?
For instance, c(something)+e(something else)+ . . .

 

[blamocur]

I see.
And what about the other question I asked specifically about the equations?

Do I rearrange 0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3-(-f+g-h+j)x^4
such that like-terms in terms of variables are grouped together, as in x(something)+x^2(something else)+ . . .
or did I NOT have to distribute the two expressions in the first place and just end up with an equation in which the coefficients are taken out?
For instance, c(something)+e(something else)+ . . .

What do you think? What are the coefficients of your polynomial (which has to be 0) ? BTW, sorry for the typo in my previous post: of course I meant "... that every coefficient is 0" instead of

... that very coefficient is 0.

 

[TheWrathOfMath]

What do you think? What are the coefficients of your polynomial (which has to be 0) ? BTW, sorry for the typo in my previous post: of course I meant "... that every coefficient is 0" instead of

The coefficients are c,e,f,g,h,j.

 

[blamocur]

The coefficients are c,e,f,g,h,j.

Those are variables used in the expressions for the coefficients. Let's be more specific: what is the coefficient, say, for [imath]x^2[/imath] ?

 

[TheWrathOfMath]

Those are variables used in the expressions for the coefficients. Let's be more specific: what is the coefficient, say, for [imath]x^2[/imath] ?

I got you.
So the equation will be of the form

0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3-(-f+g-h+j)x^4

0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3+fx^4-gx^4+hx^4-jx^4

0 = 1(-f)+x(2c-g)+x^2(c-h)+x^3(2e-j)+x^4(e+f-g+h-j)

Is that right?

 

[TheWrathOfMath]

Those are variables used in the expressions for the coefficients. Let's be more specific: what is the coefficient, say, for [imath]x^2[/imath] ?

Perhaps an easier approach would be to take 2cx+cx^2+2ex^3+ex^4 and substitute in the condition of U, so that:

2c(-1)+c(-1)^2+2e(-1)^3+e(-1)^4 =0

 

[blamocur]

I got you.
So the equation will be of the form

0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3-(-f+g-h+j)x^4

0=2cx+cx^2+2ex^3+ex^4-f-gx-hx^2-jx^3+fx^4-gx^4+hx^4-jx^4

0 = 1(-f)+x(2c-g)+x^2(c-h)+x^3(2e-j)+x^4(e+f-g+h-j)

Is that right?

Definitely a step in the right direction. But I'll rephrase my previous question: if you have a polynomial f(x) which must be 0 for all x's what can you tell about its coefficients ?

 

[blamocur]

Perhaps an easier approach would be to take 2cx+cx^2+2ex^3+ex^4 and substitute in the condition of U, so that:

2c(-1)+c(-1)^2+2e(-1)^3+e(-1)^4 =0

I haven't thought of this -- looks like you found an easier, and thus a better, way to solve it. Can you get the final answer and verify it?

 

[TheWrathOfMath]

I haven't thought of this -- looks like you found an easier, and thus a better, way to solve it. Can you get the final answer and verify it?

I got that the basis for the intersection is {2x+x^2-x^3-x^4}.
It also fits with the dimension formula, since I calculated U+W (for another part of the question), and got
dim(U+W)=5, dimU=4, dimW=2, dim(U∩W)=1.

 

[blamocur]

A agree withe dimension part, but have you tried to verify the result. I.e., does it satisfy both criteria, i.e., for U and W?