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

[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.

 

[Dick]

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.

 

[amanda_ou812]

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.

 

[Dick]

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.

 

[amanda_ou812]

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..

 

[Dick]

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.

 

[amanda_ou812]

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!

 

[HallsofIvy]

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