Finding a basis for null(T) and range(T)

[tehme1]

Homework Statement

Let Y:P₃(R) onto P₂(R) is defined by T(a₀+a₁z+a₂z²+a₃z³)=a₁+a₂z+a₃z². Find bases for null (T) and range (T). What are their dimensions?

Homework Equations

The Attempt at a Solution

Well, I am assuming the range is the a1+a2z+a3z^2, so it has dimension of 3, and basis (1, z, z^2). As for the null (T) I'm guessing it has dimension 1, since P^3 has dim = 4, and range (T) has dim 3, so 4-3 = 1.
As for the basis of null(T) I'm not sure. I'm guessing it is when a₁+a₂z+a₃z^2 = 0. But I'm not sure how you would solve it or really how you figure these out when they are polynomials and not vectors or matrices. Thanks.

 

[Mark44]

Homework Statement

Let Y:P₃(R) onto P₂(R) is defined by T(a₀+a₁z+a²z²+a₃z³)=a₁+a₂z²+a₃z². Find bases for null (T) and range (T). What are their dimensions?

Homework Equations

The Attempt at a Solution

Well, I am assuming the range is the a1+a2z+a3z^2, so it has dimension of 3

According to what you wrote, the range is two-dimensional. Was this a typo?

a₁+a₂z²+a₃z²

Every degree-three polynomial gets mapped to a polynomial that consists of a constant + a squared term. There are no terms in x or in x³ in the range.

, and basis (1, z, z^2). As for the null (T) I'm guessing it has dimension 1, since P^3 has dim = 4, and range (T) has dim 3, so 4-3 = 1.
As for the basis of null(T) I'm not sure. I'm guessing it is when a₁+a₂z+a₃z^2 = 0. But I'm not sure how you would solve it or really how you figure these out when they are polynomials and not vectors or matrices. Thanks.

 

[tehme1]

yes, it was typo. I made an edit so now the question is as it should be. Thanks

 

[Mark44]

Well, I am assuming the range is the a1+a2z+a3z^2, so it has dimension of 3, and basis (1, z, z^2). As for the null (T) I'm guessing it has dimension 1, since P^3 has dim = 4, and range (T) has dim 3, so 4-3 = 1.
As for the basis of null(T) I'm not sure. I'm guessing it is when a₁+a₂z+a₃z^2 = 0. But I'm not sure how you would solve it or really how you figure these out when they are polynomials and not vectors or matrices. Thanks.

Start by finding out what T does to the individual functions in a basis for the domain function space.
T(1) = ?
T(x) = ?
T(x²) = ?
T(x³) = ?

That might give you some understanding of what exactly gets mapped to 0 in the range.