Help with definitions in linear algebra

[devoured_elysium]

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

a) Injective
b) Bijective

Thanks in advance

 

[quasar987]

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.

 

[HallsofIvy]

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.

 

[devoured_elysium]

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

 

[quasar987]

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.