Let V and W be vector spaces and B and B' be bases respectively.Suppose T is a linear map from V to W. Then T is isomorphism if and only if T restricted to B is a bijection from B to B'.
My try is:
if T|B is a bijection then clearly T is an isomorphism. now if T is an isomorphism then it takes B to T(B) bijectively where T(B) will be a basis.
now how can we show that T|B is bijective from B to B'.
Isomorphism of vector spaces implies bijection between bases
0
$\begingroup$
linear-algebra
-
0This is not true. Suppose $V=W$ and you have two bases $B$ and $B'$ of $V$. Suppose $T=Id_V$, then $T(B)$ is not necessarily $B'$. However, you can show that $T(B)$ bijective to $B'$. – 2017-01-27
-
0There not need to be $T(B)=B'$. If $T:V \to W$ is an isomorphism, it takes basis to basis, but if you fix a basis $B$ of $V$ and a basis $B'$ of $W$, then $T$ could take $B$ to another basis $B''$ of $W$ such that $B' \neq B''.$ – 2017-01-27