# Help with definitions in linear algebra

## Main Question or Discussion Point

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

a) Injective
b) Bijective

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quasar987
Homework Helper
Gold Member
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
Homework Helper
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.

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