Help with definitions in linear algebra

  • #1

Main Question or Discussion Point

Having a matrix, how can I know if the function the matrix is representing is:

a) Injective
b) Bijective

Thanks in advance
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
say the map is L:V-->W

it is injective iff for all w in W, there is at most one v in with with L(v)=w

it is bijective if it is injective and surjective.

These are the definitions. If you're having trouble applying them to a specific problem, you should tell us what the exact problem is and what you've attempted (if anything) and then we can help more.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,793
922
Well, you could look at the definitions of those words! Any function, from one set to another is call "injective" if and only if f(x)= f(y) implies x= y: in other words, different members of the domain are mapped to different members of the range. That does NOT imply "surjective"- that something is mapped to every member of the range. For linear transformations, represented by a matrix, injective means that the domain space is mapped one-to-one to a subspace of the range space: that the rank of the matrix is equal to the dimension of the domain space, not necessairily the dimension of the range space.

In order to be "bijective" a mapping must be both injective and surjective: "one-to-one" and "onto". For a linear transformation represented by a matrix, that means it must be n by n for some positive integer n and have rank n.
 
  • #4
Thanks by your replies

If I understood well, any matrix with non-zero determinant will be bijective, right?

I don't quite get what you mean by domain space and range space. I know what a vector space is. Is it related to it?

Thanks
 
  • #5
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
If f:A-->B is a function, then A is called the domain and the subset f(A) of B is called the range. In the case where f is a linear map from one vector space to another, A is the domain space (or domain vector space) and f(A) is the range space.
 

Related Threads for: Help with definitions in linear algebra

  • Last Post
Replies
16
Views
6K
Replies
3
Views
2K
  • Last Post
Replies
4
Views
2K
Replies
17
Views
3K
Replies
7
Views
1K
Replies
9
Views
849
Top