I am doing the Khan Academy linear algebra course. I am around halfway through in which Sal talks about linear maps.

I am watching the video 'proof: Invertibility implies a unique solution to f(x)=y', in which he proves that for:


...there is only 1 unique $x$ that satisfies the equation.

He then says that because of this we can conclude that for:


...there is only 1 unique solution.

How does the first point imply the second? This is what has been stumping me.


1 Answer 1


We have that $f^-1(y)=x$ and we have to show that $f(x)=y$.

Simply substitute $f^-1(y)=x$ in $f(x)=y$ to get $f(f^-1(y))=y$. Then we get that y=y, which of course is unique since we had fixed the value of y.

  • $\begingroup$ Thanks for the help. $\endgroup$ Nov 24, 2022 at 4:48

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