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In $\mathbb{R}^4$, Let $U=span((1,1,0,0),(1,1,1,2))$. Find $u\in U$ such that $\|u-(1,2,3,4)\| $ is minimum.

I guess I need to find the basis of $U^{\perp} $, and then rewrite $(1,2,3,4)=u+u'$, where $u'\in U^{\perp}$ and $u\in U.$ Hence the minimum of $\|u-(1,2,3,4)\|$ would be $u'$. Is my argument valid?

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    Other than noting that the result would be $\|u'\|$, not $u'$: you've got it. – 2017-01-24
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    Or you could note that $u - (1,2,3,4)$ must be orthogonal to each spanning vector in U and write $u = a u_1+b u_2$ where $u_1$ and $u_2$ are the spanning vectors. – 2017-01-24

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