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Let $U = \operatorname{span}\{(1,1,-1),(2,3,-1),(3,1,-5)\}$ and $V = \operatorname{span}\{(1,1,-3),(3-2,-8),(2,1,-3)\}$. Then what is $U\cap V$?

My attempt: there is little I can do, because I do not know how to calculate intersection. I can determine the span of both, but how to compute intersection.

P.S : please dont reduce it to a linear system to solve and determine, because, it can only work for lower dimensional spaces. I want a general method.

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You can determine $U_1=U^{\perp}$ and $V_1=V^{\perp}$; then $$ U\cap V=U_1^\perp\cap V_1^{\perp}=(U_1+V_1)^\perp $$ Now, using column vectors, $U_1$ is the null space of the matrix $$ \begin{bmatrix} 1 & 1 & -1 \\ 2 & 3 & -1 \\ 3 & 1 & -5 \end{bmatrix} $$ A row reduction brings the matrix in the form $$ \begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$ so $U_1$ is the span of $\begin{bmatrix}2\\-1\\1\end{bmatrix}$

You find similarly $V_1$. Then you can find a basis for $U_1+V_1$ and determine a basis for its orthogonal complement.

In this case it's much easier, though, because $V$ is quite simple to describe.

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    Hello, what do u mean by orthogonal? And orthogonal compliment? – 2017-01-26
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    @Shobhit If you don't know those concepts, I'm afraid you can't get really general methods. – 2017-01-26
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    Are there definitions elementary? Should i wiki it, will i be able to grasp concepts of it? P.S i am an undergraduate ( 2nd year) – 2017-01-26
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    Also, if not the general method, how else can i do it? – 2017-01-26
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    @Shobhit First of all, determine the dimension of the spaces involved, then you'll have a clear idea of what's happening. At least you can easily determine the dimension of $U$, $V$ and $U+V$, so also the dimension of the intersection, via Grassmann's formula. But, in this case, there's not much to do, really: compute the dimensions and you'll see. – 2017-01-26
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    Ok, i will try and let u know. Thank u :) – 2017-01-26
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    I am a fan of your answers Dr. (egreg? I really don't know whether this is your real name). Always enlightening and concise. – 2017-01-26
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    @OpenBall Thanks. No, it's not my real name, but you can easily find it by looking carefully in my profile page. – 2017-01-26
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    Ok i was able to determine the dimensions of U and V as 2,3 respectively. How to determine U+V, and its dimension? Also even after determining the dimension of U $/intersection$ V, i cannot match it with the given options. The given options are a) U, b)V, c) the {0} subspace, d) none. Sorry for bothering you again, these type of questions will be on my test, so little worried. – 2017-01-26
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    @Shobhit Since $\dim V=3$, what's it? And then, what's the intersection? – 2017-01-26
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    Ohhhhhhh, since dim is 3 its R^3 and therefore intersection is U. Thank u soooo much :) – 2017-01-26
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    How to determine U+V? – 2017-01-26
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    @Shobhit In this case it's not needed, because of the fact that $V$ is the whole space. In general, you just put together the spanning sets. – 2017-01-26
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    I got the answer, but i was just wondering. "Put together the spanning set", what does it mean? – 2017-01-26
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    @Shobhit If $U=\operatorname{span}\{u_1,\dots,u_m\}$ and $V=\operatorname{span}\{v_1,\dots,v_n\}$, then $U+V=\operatorname{span}\{u_1,\dots,u_m,v_1,\dots,v_n\}$ – 2017-01-26
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    Oh ok. Thank u again – 2017-01-26