Linear Operator Rules for Ker T and Ker S: Solving for T and S in R^3

In summary, the conversation discusses finding the rule for the linear operator S such that ker S is equal to ker T and I am S is equal to ker T. It is mentioned that S may be assumed to be the inverse operator of T, but it is not clear if this is the case. The conversation also mentions finding ker T and I am T as a first step, but the individual is still unsure how to proceed. They suggest describing the operators geometrically to gain a better understanding.
  • #1
Cyannaca
22
0
Let T: R^3 -> R^3 be a linear operator and
T(x;y;z)= (x -y + 3z;2x-y+8z;3x -5y+5z).

Find the rule, for the linear operator S: R^3 -> R^3 such that ker S= I am T and I am S=Ker T.

I'm not really sure how I should start this problem. Also I would like to know if I can assume S is the inverse operator of T.
 
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  • #2
Seems obvious to me that the first thing to do is to find ker T and I am T. *shrug*

Also I would like to know if I can assume S is the inverse operator of T.

What do you think?
 
  • #3
" Also I would like to know if I can assume S is the inverse operator of T."

If the two kernals are not trivial, then T does not HAVE an inverse!
 
  • #4
I found ker T and I am T and I'm still stuck. I know it's not the inverse operator but I still don't know how to find the rule for the linear operator S.
 
  • #5
Maybe if you described them geometrically, it would spark an idea?
 

1. What is a linear operator?

A linear operator is a mathematical function that maps one vector space to another, while preserving the structure of the vector space. In simpler terms, it is a function that takes in a vector and outputs another vector, where the vector space operations of addition and scalar multiplication are preserved.

2. What is the purpose of a linear operator?

A linear operator is used to describe and analyze linear transformations. It is also a fundamental tool in solving mathematical problems, particularly in the field of linear algebra.

3. How do you solve a problem involving linear operators?

To solve a problem involving linear operators, it is important to understand the properties of linear operators and how they affect vectors. This includes understanding the concept of linearity, as well as properties such as commutativity and associativity. Then, you can apply mathematical techniques such as linear algebra and matrix operations to manipulate and solve the problem.

4. What are some common applications of linear operators?

Linear operators have many applications in various fields, such as physics, engineering, and computer science. Some examples include solving differential equations, analyzing dynamical systems, and creating algorithms for data processing and machine learning.

5. Can linear operators be used for non-linear transformations?

No, by definition, linear operators preserve the structure of the vector space and therefore cannot be used for non-linear transformations. However, they can be combined with other mathematical tools to approximate or model non-linear transformations.

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