# Math Help - Showing two subspaces are a direct sum

1. ## Showing two subspaces are a direct sum

Consider the subspaces S1={(r,0,t): r,t∈R} and S2={(s,s,0): s∈R} of R^3. Prove that S1⊕S2=R^3.

I am not sure how to show this at all, my teacher didn't give us any examples of this. Would really appreciate some help please.

2. ## Re: Showing two subspaces are a direct sum

Originally Posted by steph3824
Consider the subspaces S1={(r,0,t): r,t∈R} and S2={(s,s,0): s∈R} of R^3. Prove that S1⊕S2=R^3.

I am not sure how to show this at all, my teacher didn't give us any examples of this. Would really appreciate some help please.
Prove that every vector $(a,b,c)\in\mathbb{R}^3$ can be written as sum of vector from $S_1$ and vector from $S_2$ in an unique way.

And it's clear that $S_1\cap S_2=\{0\}$

These two things will give you $S_1\oplus S_2 =\mathbb{R}^3$