Vector spaces and direct sums

Question

The map that was constructed in lectures is:

$V,W$ subspaces of $U$. $f\colon V \oplus W \to U$ by the formula: $f((v,w))=v+w$ for $v$ in $V$, $w$ in $W$

Is it correct to generalise this to, say, $f((v_1, v_2,\dotsc, v_n))=v_1+v_2+\dotsb+v_n$?

How do I show that the map I have constructed is an isomorphism for the given criterion?

Answer 1

1. It respects operation (obvious).

2. It is surjective (easy to show).

3. Assume $w = (v_1, \dots, v_n)$ that $f(w) = 0$. Okay, let's write $f(w) = v_1 + \dots v_n = 0$. Then $v_i = 0, \forall i \in \{1, \dots, n\}$ because $V_i \cap (V_1 + \dots + V_{i-1} + V_{i+1} + \dots + V_n) = \{0\}$. From this we got that $w = 0$ and $f$ injective.