Showing two subspaces are a direct sum

• Jul 14th 2011, 01:58 PM
steph3824
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.
• Jul 14th 2011, 02:46 PM
Also sprach Zarathustra
Re: Showing two subspaces are a direct sum
Quote:

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 $\displaystyle (a,b,c)\in\mathbb{R}^3$ can be written as sum of vector from $\displaystyle S_1$ and vector from $\displaystyle S_2$ in an unique way.

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

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