How Do You Determine if a Linear Transformation is Injective?

[mbud]

I was just wondering how you know if linear transformations injective?

 

[matt grime]

You work out its kernel. If it's a map from V to V then you can work out its determinant which tells you if the kernel is zero or not (but not what it is).

 

[mbud]

thanks

 

[HallsofIvy]

A function, f, in general, is injective if f(x)= f(y) implies x= y. If f is linear, then f(x)= f(y) gives f(x)- f(y)= f(x-y)= 0 while x= y is the same as x- y= 0. That is why a linear function is injective if and only if its kernel is trivial: {0}, as matt grime said.