Proofs from Matrix representations All proofs from Matrix representations.
[ PROOF ] Write
v = c 1 v 1 + ⋯ + c n v n . v=c_1v_1+\cdots+c_nv_n. v = c 1 v 1 + ⋯ + c n v n . Then
[ v ] B V = ( c 1 ⋮ c n ) . [v]_{B_V}=
\begin{pmatrix}
c_1\\
\vdots\\
c_n
\end{pmatrix}. [ v ] B V = c 1 ⋮ c n . Preservation of finite linear combinations gives
[ T ( v ) ] B W = c 1 [ T ( v 1 ) ] B W + ⋯ + c n [ T ( v n ) ] B W = c 1 a 1 + ⋯ + c n a n = A [ v ] B V . \begin{aligned}
[T(v)]_{B_W}
&=c_1[T(v_1)]_{B_W}+\cdots+c_n[T(v_n)]_{B_W}\\
&=c_1a_1+\cdots+c_na_n\\
&=A[v]_{B_V}.
\end{aligned} [ T ( v ) ] B W = c 1 [ T ( v 1 ) ] B W + ⋯ + c n [ T ( v n ) ] B W = c 1 a 1 + ⋯ + c n a n = A [ v ] B V . This proves the coordinate identity. If another matrix has the same identity, apply it to each v j v_j v j [ v j ] B V [v_j]_{B_V} [ v j ] B V j j j j j j [ T ( v j ) ] B W = a j [T(v_j)]_{B_W}=a_j [ T ( v j ) ] B W = a j A A A
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