i was reading my notes and it states that the kernel of a function allows you to check if the function is injective.

how do you check with a kernel? isint elements in the kernel, elements that through the function get mapped to the identity?

What are we "in" here? Linear maps? Group homomorphisms?

The proof is pretty much the same in every context through.

Let \(\displaystyle T\) be a linear map.

Then \(\displaystyle T\) is injective

\(\displaystyle \Leftrightarrow (Tx = Ty \Rightarrow x=y) \)

\(\displaystyle \Leftrightarrow (Tx-Ty = 0 \Rightarrow x=y) \)

\(\displaystyle \Leftrightarrow (T(x-y) = 0 \Rightarrow x = y) \)

\(\displaystyle \Leftrightarrow (T(x-y) = 0 \Rightarrow x-y = 0) \)

\(\displaystyle \Leftrightarrow (Tu = 0 \Rightarrow u =0) \)

\(\displaystyle \Leftrightarrow kerT = \{0\}\).