Basis for kernel of linear transform

In summary, you can find the kernel and range of L by looking at the equation that defines it, and by finding all the polynomials that satisfy the equation.
  • #1
Takuya
4
0
Hey guys!

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

L is a linear transform that maps L:P4[tex]\rightarrow[/tex]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?
 
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  • #2
Takuya said:
Hey guys!

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

L is a linear transform that maps L:P4[tex]\rightarrow[/tex]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 [itex]L[/itex]. What relationships must hold among [itex]a_1, a_2, a_3, a_4[/itex] in order to obtain

[tex]L(a_1 t^3 + a_2 t^2 + a_3 t + a_4) = 0[/tex]

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

[tex](a_1 - a_2) t^3 + (a_3 - a_4) t[/tex]
 
  • #3
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} ?
 
  • #4
Takuya said:
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 [itex]a (1+t) + b(t^2 + t^3)[/itex].

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].
 
  • #5
Ohhh right right. Yeah I was getting ahead of myself. I understand now! Thanks :)
 

1. What is the basis for the kernel of a linear transform?

The basis for the kernel of a linear transform is the set of all vectors that are mapped to the zero vector by the transform. In other words, it is the set of all inputs that result in an output of zero.

2. Why is the kernel important in linear algebra?

The kernel is important because it allows us to understand the behavior of a linear transform. It helps us identify which inputs will result in a zero output and which inputs will not be affected by the transform.

3. How is the basis for the kernel determined?

The basis for the kernel is determined by solving the system of linear equations represented by the linear transform. The linear equations are set to equal zero and then solved to find the values of the variables that result in a zero output.

4. What is the relationship between the kernel and the range of a linear transform?

The kernel and the range of a linear transform are complementary. The basis for the kernel represents the inputs that result in a zero output, while the basis for the range represents the outputs that can be reached by the transform.

5. Can the basis for the kernel be empty?

Yes, it is possible for the basis for the kernel to be empty. This means that there are no inputs that result in a zero output for the linear transform. In this case, the transform is known as a one-to-one transform because each input has a unique output.

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