Kernal, range and linear transformations

[war485]

Homework Statement

T: P₂ --> P₂ be a linear transformation defined by T(p(x)) = xp'(x)

where ' is the derivative
Describe the kernal and range of T and are any of the following polynomials in the range and or in the kernal of T?
2
x²
1 - x

Homework Equations

power rule (for derivatives)

The Attempt at a Solution

I took an arbitrary a+bx+cx² and turned it into bx + 2cx² after applying T on it. I understand that the kernal means some vector that will turn it into the zero vector (like finding its null). so does that b = c = 0 in order for this to work? Then will it imply that the kernal of T is the set of all constant polynomials? If this is right, then from this I think x² and 2 and 1 - x should be in ker(T).

As for the range of T, I think it is all of P², so that includes the 2, x² and 1 - x.

Is all of what I've just done correct? I like checking my work here when I get stuck, you guys are so nice and great at helping people! :)

 

[Mark44]

Homework Statement

T: P₂ --> P₂ be a linear transformation defined by T(p(x)) = xp'(x)

where ' is the derivative
Describe the kernal and range of T and are any of the following polynomials in the range and or in the kernal of T?
2
x²
1 - x

Homework Equations

power rule (for derivatives)

The Attempt at a Solution

I took an arbitrary a+bx+cx² and turned it into bx + 2cx² after applying T on it.

Or more mathematically, T(a+bx+cx²) = bx + 2cx².
To find the kernel (no such word as kernal) of T, what values of a, b, and c give you output polynomials that are zero for any value of x? IOW, for what values of a, b, and c is bx + 2cx² identically zero?

I understand that the kernal means some vector that will turn it into the zero vector (like finding its null). so does that b = c = 0 in order for this to work? Then will it imply that the kernal of T is the set of all constant polynomials? If this is right, then from this I think x² and 2 and 1 - x should be in ker(T).

Constant polynomials are in the kernel, but the other two functions aren't.

As for the range of T, I think it is all of P², so that includes the 2, x² and 1 - x.

No, there are some polynomials in P² that aren't in the range of T. Look at this equation again--T(a+bx+cx²) = bx + 2cx²--and notice that there are some polynomials that aren't in the range.

Is all of what I've just done correct? I like checking my work here when I get stuck, you guys are so nice and great at helping people! :)

 

[HallsofIvy]

Homework Statement

T: P₂ --> P₂ be a linear transformation defined by T(p(x)) = xp'(x)

where ' is the derivative
Describe the kernal and range of T and are any of the following polynomials in the range and or in the kernal of T?
2
x²
1 - x

Homework Equations

power rule (for derivatives)

The Attempt at a Solution

I took an arbitrary a+bx+cx² and turned it into bx + 2cx² after applying T on it. I understand that the kernal means some vector that will turn it into the zero vector (like finding its null). so does that b = c = 0 in order for this to work? Then will it imply that the kernal of T is the set of all constant polynomials?

Okay

If this is right, then from this I think x² and 2 and 1 - x should be in ker(T).

Are you saying that you think x² and 1- x are constant polynomials?

As for the range of T, I think it is all of P², so that includes the 2, x² and 1 - x.

You said above that T(a+ bx+ cx²)= bx+ 2cx². How do you get 1- x from that?

Is all of what I've just done correct? I like checking my work here when I get stuck, you guys are so nice and great at helping people! :)

I'm not! I'm mean and grumpy!

 

[war485]

oups sorry for misspelling kernel.

I looked up constant polynomial again and realized I made another mistake... only 2 is a constant polynomial in this case. So only 2 should be in ker(T). My bad...

And so, I think the range should be something like the span of these basis: x and x² ? I realized I couldn't get 1 - x in there, so x² should be in the range of T.
Thanks for pointing that out.

HallsOfIvy, I see you as a nice person too, not grumpy and not mean, you're still helping me just like 2 months ago or so (can't remember exactly) but I can see why you're upset about me originally thinking about what constant polynomial was... :)

Did I get it right this time?

 

[Mark44]

oups sorry for misspelling kernel.

I looked up constant polynomial again and realized I made another mistake... only 2 is a constant polynomial in this case. So only 2 should be in ker(T). My bad...

You're not saying that 2 is the only constant polynomial in the kernel of T, are you?

And so, I think the range should be something like the span of these basis: x and x² ? I realized I couldn't get 1 - x in there, so x² should be in the range of T.
Thanks for pointing that out.

HallsOfIvy, I see you as a nice person too, not grumpy and not mean, you're still helping me just like 2 months ago or so (can't remember exactly) but I can see why you're upset about me originally thinking about what constant polynomial was... :)

Did I get it right this time?

 

[war485]

no, 2 is not the only constant, but the question also asked me if 2 was included in there. Ker(T) is all constant values.

*edit* I think I got it. Thanks Mark, and Ivy.