# Finding a Linear Transformation T: R2 -> R2 with Equal Null Space and Range

• amanda_ou812
In summary, a linear transformation T: R2 -> R2 can be defined such that the null space is equal to the range by choosing a one dimensional subspace, such as y=x or x=0, and mapping any vector in that subspace into zero. Examples of such transformations include T(x,y) = (x-y, x-y) or A(x,y) = (y, 0).
amanda_ou812

## Homework Statement

Give an example of a linear transformation T: R2 -> R2 such that the null space is equal to the range.

## Homework Equations

null space and range

## The Attempt at a Solution

I have been trying to come up with a solution but I cannot figure it out. What might be a logical methodology.

Well, you know the dimension of the null space plus the dimension of the range must be 2 by the rank nullity theorem. If you want them to be the same subspace, what should the dimension be? That's a good place to start.

well dim V is 2 so nullity and rank have to each be 1. I know that N(T) is a subset of V and R(T) is a subset of W so if R(T) = N(T) = a set with one vector and that vector will belog to both V and W...i guess that thought process does not get me closer to a conclusion.

amanda_ou812 said:
well dim V is 2 so nullity and rank have to each be 1. I know that N(T) is a subset of V and R(T) is a subset of W so if R(T) = N(T) = a set with one vector and that vector will belog to both V and W...

Ok, so pick anyone dimensional subspace of R^2. What's your favorite? Now cook up a linear transformation that maps any vector into that one dimensional subspace and any vector in that subspace into zero. It helps to be concrete about what the subspace is.

I am not really sure if this is what you mean...
so, I choose y=x as my one dim subspace of R2. So I could do T(x, y) = (x,x).
but what do you mean by mapping any vector in this subspace into zero?
If I use this transformation my R(T) = (1, 1) and my N(T) = c (0, 1)...I feel like I am close to getting the answer..

amanda_ou812 said:
I am not really sure if this is what you mean...
so, I choose y=x as my one dim subspace of R2. So I could do T(x, y) = (x,x).
but what do you mean by mapping any vector in this subspace into zero?
If I use this transformation my R(T) = (1, 1) and my N(T) = c (0, 1)...I feel like I am close to getting the answer..

There's not just one answer. You are getting close. Here's BIG hint: suppose you pick T(x,y)=(x-y,x-y)? What are the range and null space? Now find another answer, my favorite subspace is x=0. Find a T that works with that.

Last edited:

ahhh... I see now. N(T) = {(x, y) : x=y} or {c (1, 1)} for C in R and R(T) (which is all the outputs that are the same as the inputs - is how I like to think of it) will be (a1, a1) = c (1, 1) geez

I kept trying (x-y, 0) but i never thought to do (x-y, x-y) thanks!

Good! And now that you have that, here's another, simpler example: A(x, y)= (y, 0).

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another, preserving the basic algebraic structure of the space. In simpler terms, it is a function that takes in vectors and outputs new vectors, while maintaining certain properties such as linearity and preservation of the origin.

## 2. What is the null space of a linear transformation?

The null space of a linear transformation is the set of all vectors that are mapped to the zero vector in the output space. In other words, it is the set of all vectors that are "ignored" or "flattened" by the transformation.

## 3. What is the range of a linear transformation?

The range of a linear transformation is the set of all possible output vectors that can be produced by the transformation. It is also known as the image of the transformation.

## 4. How can a linear transformation have equal null space and range?

In order for a linear transformation to have equal null space and range, the transformation must be a projection. This means that the transformation "projects" all vectors onto a single line or plane, resulting in both the null space and range being the same line or plane.

## 5. Why is finding a linear transformation with equal null space and range important?

Finding a linear transformation with equal null space and range can be useful in simplifying calculations and understanding relationships between different vector spaces. It can also be used in applications such as image processing, where projections are commonly used to manipulate images.

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