Basis for kernel of linear transform

[Takuya]

Hey guys!

I am having a major brain problem today, with this problem.

L is a linear transform that maps L:P₄$$\rightarrow$$P₄

As such that (a₁t³+a₂t²+a₃t+a₄ = (a₁-a₂)t³+(a₃-a₄)t.

I am trying to find the basis for the kernel and range.

I know that the standard basis for P₄ is {1,x,x²,x³}
And the kernel is when L(u)=0, but I don't know how to find the transformation matrix, since we're not dealing with numbers in R, but in the set of polynomials. Is there another way to find the kernel/range and bases without using the T matrix?

 

[jbunniii]

Hey guys!

I am having a major brain problem today, with this problem.

L is a linear transform that maps L:P₄$$\rightarrow$$P₄

As such that (a₁t³+a₂t²+a₃t+a₄ = (a₁-a₂)t³+(a₃-a₄)t.

I am trying to find the basis for the kernel and range.

I know that the standard basis for P₄ is {1,x,x²,x³}
And the kernel is when L(u)=0, but I don't know how to find the transformation matrix, since we're not dealing with numbers in R, but in the set of polynomials. Is there another way to find the kernel/range and bases without using the T matrix?

Don't worry about trying to find a matrix. It's not necessary in order to answer this problem.

To find the kernel, just look at the defining equation for $L$. What relationships must hold among $a_1, a_2, a_3, a_4$ in order to obtain

$$L(a_1 t^3 + a_2 t^2 + a_3 t + a_4) = 0$$

To find the range, simply answer this: if you allow $a_1, a_2, a_3, a_4$ to take on all possible values, what are all the possible polynomials that you can produce of the form

$$(a_1 - a_2) t^3 + (a_3 - a_4) t$$

 

[Takuya]

a₁=a₂ and a₃=a₄, so that a₂ and a₄ can be arbitrary. So plugging back into the original L(a), and factoring, you'd get that {(1+t),(t²+t³)} is the kernel of L?

The range would be … any value of a₂ or a₄ for t³ and t, so the range is just {t,t³} ?

 

[jbunniii]

a₁=a₂ and a₃=a₄, so that a₂ and a₄ can be arbitrary. So plugging back into the original L(a), and factoring, you'd get that {(1+t),(t²+t³)} is the kernel of L?

That is a BASIS for the kernel of L. The kernel itself consists of all possible linear combinations of the basis elements, which is any polynomial of the form $a (1+t) + b(t^2 + t^3)$.

The range would be … any value of a2 or a4 for t3 and t, so the range is just {t,t3} ?

That is a BASIS for the range of L. The range itself is once again the set of all possible linear combinations of the basis elements, i.e. anything of the form [itex]a t + b t^3[/tex].[/itex]

 

[Takuya]

Ohhh right right. Yeah I was getting ahead of myself. I understand now! Thanks :)