I am having trouble proving the following statement:
Let V be a n-dimensional vector space with an ordered basis $\displaystyle \beta$. Prove that $\displaystyle T:V \rightarrow F^n$ by $\displaystyle T(x) = [T]_{\beta}$ is linear.
I am having trouble proving the following statement:
Let V be a n-dimensional vector space with an ordered basis $\displaystyle \beta$. Prove that $\displaystyle T:V \rightarrow F^n$ by $\displaystyle T(x) = [T]_{\beta}$ is linear.
So far I have:
$\displaystyle \beta$= {$\displaystyle u_1,u_2,...,u_n$} and $\displaystyle a_1,a_2,...,a_n$ are unique scalars.
$\displaystyle x = \sum ^{n}_{i=1}a_{i}u_{i}$
$\displaystyle T(x) = T(\sum ^{n}_{i=1}a_{i}u_{i}) = \sum ^{n}_{i=1}a_{i}T(u_{i})$
I'm not sure if this is correct.