Proving the Existence of Direct Sums in Linear Algebra

[omoplata]

In 'Linear Algebra Done Right' by Sheldon Axler, a direct sum is defined the following way,

We say that V is the direct sum of subspaces U_1, \dotsc ,U_m written V = U_1 \oplus \dotsc \oplus U_m, if each element of V can be written uniquely as a sum u_1 + \dotsc + u_m, where each u_j \in U_j.

Suppose V = U \oplus W. Is there any way I can prove that for all u \in U there exists v \in V and w \in W such that v = u + w?

If that can be done, then I can solve a problem given later in the book.

 

[micromass]

Take w=0 and v=u.

 

[omoplata]

LOL, OF COURSE!

Thanks!