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Math Help - Showing two subspaces are a direct sum

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    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.
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    Re: Showing two subspaces are a direct sum

    Quote Originally Posted by steph3824 View Post
    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
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