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

  • Context: Undergrad 
  • Thread starter Thread starter squenshl
  • Start date Start date
Click For Summary
SUMMARY

The discussion focuses on calculating the dimensions of the range and kernel of a linear transformation T, specifically when T is one-to-one and onto. According to the rank-nullity theorem, for a linear transformation from a vector space of dimension n to a vector space of dimension m, the equation dim(Ran(T)) + dim(Ker(T)) = n holds true. When T is one-to-one, it follows that dim(Ker(T)) equals 0, while if T is onto, dim(Ran(T)) equals m. The terms "rank" and "nullity" are also defined in this context.

PREREQUISITES
  • Understanding of linear transformations
  • Familiarity with vector spaces
  • Knowledge of the rank-nullity theorem
  • Basic concepts of isomorphisms
NEXT STEPS
  • Study the rank-nullity theorem in detail
  • Explore examples of one-to-one and onto linear transformations
  • Learn about isomorphisms in linear algebra
  • Investigate the implications of kernel and range in vector spaces
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching concepts related to linear transformations and vector spaces.

squenshl
Messages
468
Reaction score
4
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?
 
Physics news on Phys.org
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.
 
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.
 

Similar threads

Graduate How Does the Non-Equality of Kernels Imply Their Sum Equals the Vector Space?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Show that dim(Ker(SoT))<=dim(KerT)+dim(KerS)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Proof that two linear forms kernels are equal
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Understanding proof for theorem about dimension of kernel
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Given two linear transformations L and K, show ##K = \lambda L## holds
  • · Replies 9 ·
Replies
9
Views
2K
MHB Kernel of Linear Map $\theta$ in $\mathbb{F}_{q^n}$
  • · Replies 24 ·
Replies
24
Views
4K
Undergrad Notation questions about kernel, cokernel, image and coimage
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad Trying to get a better understanding of the quotient V/U in linear algebra
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
3K