$E$ and $E'$ are finite dimension vector spaces.
I need to prove that if $f$ is a linear mapping from $E$ to $E'$ then it is a bijective function if and only if the mappings from a basis of $E$ using f generate a basis of $E'$.
Assuming $f$ is a bijective function, the mapped vectors using f are both linearly independent and span $E'$. How can I prove that $dimE$ = $dimE'$? I mean, I need to show that the correspondence one-to-one generates basis of $E'$ using the basis vectors of $E$.
Best Answer
You cannot use dimensions since the spaces are not given to be finite dimensional. Let $(x_i)$ be basis for $E$ and $y \in E'$. Since $f$ is surjective there exists $x \in E$ such that $y=f(x)$. Now write $x$ as a linear combination of $x_i$'s and use linearilty of $f$ to conclude that $y=f(x)$ is a linear combination of $f(x_i)$'s. This proves that $(f(x_i))$ spans $E'$. Since they are also linearly independent they form a basis for $E'$.