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Let $\mathcal V$ be the subspace $\mathcal span \ \mathbf x$, where $\mathbf x$=[3 2 1]$^T$. Find the best approximation in $\mathcal V$ $\mathcal to \ \mathbf y$ =[1 2 3]$^T$ with respect to the following norms:

A. $\Vert\cdot\Vert_1$

B. $\Vert\cdot\Vert_2$

C. $\Vert\cdot\Vert_\infty$

Any help is truly appreciated, as I have no idea where to begin. Does it have something to do with the physical distance between the two vectors?

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    The distance is measured by these norms. Notice that $V =\{ t x \mid t\in \mathbb R\}$. So you are looking for the minimum of $t\to \| tx - y \|$. Notice that the 1 norm and $\infty$ norm aren't differentiale. But the resulting objective functions are just piecewise linear functions. The Euclidean norm is fairly standard ... – 2017-02-07
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    For A: $\|rx-y\|_1=|3r-1|+|2(r-2)|+|r-3|.$ Separate this into cases: (a) $\;r\geq 3,\;$ (b) $\; r\in [2,3),\;$ (c) $\;r\in [1/3,2),\;$ (d) $\; r<1/3.\;$ Find the best $r $ for each case, and take the best of the best. – 2017-02-07
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    Is the best r the smallest or largest? – 2017-02-11

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