For finite dimensional vector spaces $W_1, …, W_n$, Is it true that the theorem
$$\dim(W_1 \oplus … \oplus W_n) = \dim(W_1) + … +\dim(W_n)$$
is in fact equivalent to the Rank-Nullity theorem? I.e. It's just another way of stating the Rank-Nullity theorem?
I tried to prove the direct sum theorem by first proving $\dim(W_1 \oplus W_2) = \dim(W_1) + \dim(W_2)$. This equation seems to have a similar form as the Rank-Nullity theorem but I cannot seem to draw the link between the RHS of the above equation with the dimension of $\ker(T)$ and $\text{Im}(T)$ of linear transformation $T$ mapping from $W_1 \oplus W_2$ to itself.
Best Answer
A general element $v$ of $W_1\oplus W_2$ has unique representation as $w_1+w_2$ with $w_i$ in $W_i$.
Map $W_1\oplus W_2$ into $W_1$ by $v\mapsto w_1$. This has Rank $=\dim(W_1)$ and Nullity $=\dim(W_2)$ precsely as you want.