Showing null space and range are invariant

In summary: T(S(x)) is in ?.In summary, we need to show that for any vector x in the null space of T, S(x) is also in the null space of T, and for any vector y in the range of T, S(y) is also in the range of T. This will prove that the null space and range of T are invariant under S.
  • #1
adottree
4
0
If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.

I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?
 
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  • #2
adottree said:
If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.

I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?

Showing that it stays in V is not quite what you want to do. You want to show that every element in the range of T remains in the range of T when acted upon by S.

So say, t is an element in the range of T, what can you say about this element?
 
  • #3
adottree said:
If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.

I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?
If x is in the null space of T, then T(x)= 0. Therefore, ST(x)= ?. And so T(S(x))= ?

If y is in the range of T, then there exist x such that T(x)= y. So S(T(x))= S(y). But that is equal to T(S(x)). So S(y) is in ?
 

FAQ: Showing null space and range are invariant

Question 1: What is the null space of a matrix?

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, equal the zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the given matrix and x is a vector of appropriate size.

Question 2: What is the range of a matrix?

The range of a matrix is the set of all possible outputs that can be obtained by multiplying the matrix by any given input vector. In other words, it is the set of all possible solutions to the equation Ax = b, where A is the given matrix and b is a vector of appropriate size.

Question 3: Why is it important to show that null space and range are invariant?

Showing that null space and range are invariant means that they do not change when the matrix is transformed through operations such as row reduction or multiplication by another matrix. This is important because it allows us to make predictions and solve equations using a simpler form of the matrix without changing the solutions.

Question 4: How can we show that null space and range are invariant?

We can show that null space and range are invariant by using mathematical proofs. One method is to show that the dimensions of the null space and range remain the same after the matrix has been transformed. Another method is to show that the solutions to the equations Ax = 0 and Ax = b remain the same after the matrix has been transformed.

Question 5: What are some practical applications of understanding the invariance of null space and range?

Understanding the invariance of null space and range can be useful in many areas of science and engineering. For example, in linear algebra, it allows us to simplify calculations and solve equations more easily. In machine learning, it can help us analyze and interpret data more accurately. In physics, it can aid in understanding the behavior of physical systems. Overall, understanding this concept can help us make more accurate predictions and improve our understanding of the world around us.

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