Calculating Dim(ran(T) & Dim(Ker(T)

  • Thread starter squenshl
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In summary, to calculate Dim(Ran(T)) when T is 1-to-1, you can use the rank-nullity theorem and the fact that an isomorphism maps only the 0 vector to the 0 vector. If T is onto, then dim(Ran(T)) is equal to the dimension of the vector space T is mapping to.
  • #1
squenshl
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I have a problem.
Calculate Dim(Ran(T)) if T is 1-to-1. Also calculate Dim(Ker(T)) if T is onto.
How do you think I should do this?
 
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  • #2
There is nothing to calculate. You just need to think about it. A 1-1 map is an isomorphism onto its image for example - plus there are the standard results like the rank nullity theorem to help you.
 
  • #3
If T is a linear transformation from a vector space of dimension n to a vector space of dimension m, then dim(Ran(T))+ dim(Kernel(T))= n. That's the "rank-nullity" theorem matt grime mentioned. If T is "one-to-one", then it maps only the 0 vector to the 0 vector so dim(Kernel(T))= ? If T is "onto" what is dim(Ran(T)).

dim(Ran(T)) is also called the "rank" of T and dim(Kernel(T)) is the "nullity" of T.
 

FAQ: Calculating Dim(ran(T) & Dim(Ker(T)

What is the purpose of calculating Dim(ran(T)) and Dim(Ker(T))?

The purpose of calculating the dimensions of the range and kernel of a linear transformation (T) is to gain a better understanding of its properties and to potentially solve problems related to the transformation.

How do you calculate Dim(ran(T)) and Dim(Ker(T))?

To calculate the dimension of the range (ran(T)), you can use the rank-nullity theorem which states that the dimension of the range is equal to the rank of the transformation. The dimension of the kernel (Ker(T)) can be found by subtracting the dimension of the range from the total dimension of the vector space.

What does the dimension of the range and kernel tell us about the linear transformation?

The dimension of the range tells us the maximum number of linearly independent vectors that can be produced by the transformation. The dimension of the kernel tells us the number of free variables in the transformation, which can be used to solve systems of linear equations.

How can calculating Dim(ran(T)) and Dim(Ker(T)) be useful in solving problems?

Knowing the dimensions of the range and kernel can help determine if a linear transformation is one-to-one or onto, which can be useful in determining if a solution to a system of equations exists. It can also be used to find a basis for the range and kernel, which can be helpful in solving for a specific vector or inverting the transformation.

Are there any limitations to calculating Dim(ran(T)) and Dim(Ker(T))?

Calculating the dimensions of the range and kernel may not always be feasible, especially for larger vector spaces or more complex transformations. In these cases, alternative methods may need to be used to gain insights into the properties of the transformation.

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