1. ## Optimal Approximation

Hello,

1) Let $v \in \mathbb{R}^d$ be a unit vector. What is the meaning of the linear mapping $A:\mathbb{R}^d \rightarrow \mathbb{R}^d$, defined as

$A u = vv^T u$

2) Which linear subspace is the best approximation to the set of $n$ points in the sense that the Euclidean distance to the subspace is minimized:

$v^{*} = argmin_{v \in \mathbb{R}^d , \Vert v \Vert = 1} (\frac{1}{n} \sum_{i=1}^d \Vert X_i - (vv^T)X_i \Vert^2)$
Thank you!

2. 1) The mapping A is the orthogonal projection onto the one-dimensional subspace spanned by the unit vector v.

2) So the unit vector v that minimises the expression $\frac1n\sum_{i=1}^d\|X_i - vv^{\textsc t}X_i\|^2$ spans a line (=one-dimensional subspace) that is the best approximation to the set of $d$ points $X_1,\ldots,X_d$ in the sense that the mean of the squares of the Euclidean distances from the points to the line is minimised.

3. Thanks! I am now trying to find an expression for such a v...