Now, assume we have $2$ users, $3$ movies, and a $2$ by $3$ matrix $Y$ given by

$Y=\left[\begin{array}{ccc}1& 8& ?\\ 2& ?& 5\end{array}\right]$

Our goal is to find $U$ and $V$ such that $x=U{V}^T$ closely approximates the observed ratings in $Y.$ We want

also to encourage $U,V$ to be 'simple' in the sense that their entries are small; we quantify the strength of this

encouragement by a parameter $,$ so that our optimal solution $(U,V)$ depends on $.($BEGIN SIDENOTE: This

encouragement resolves cases that are strictly ambiguous $($for example, in

$\left[\begin{array}{ccc}?& ?& 16\\ 2& 3& ?\end{array}\right]$

that $16$ could be a product of $44$ or a product $(0\mathrm{.}00004)(400000)$; we regard the former as more

parsimonious, unless special domain knowledge suggests otherwise$).$ Moreover, this encouragement slightly

dampens our $U$ and $V$ entries toward zero, even in non$-$ambiguous cases. It's like a TA tells you they enjoy the

movie The Shining. If the TA was a true psychological horror fan, that would match the data perfectly. But

maybe the TA just happened to like that movie's particulars. So there are multiple explanations. Seeing data

increases our belief in that data's explanation$($s$)$; but when there are multiple explanations for the data, we

spread out that increase over the multiple explanations. So we don't increase our belief in a particular

explanation as much as we would increase it if other explanations were unavailable. In the case of unknown $U$

and $V,$ evidence that they approximately multiply to be some known $Y$ does not mean we should guess

values of $U,V$ that exactly multiply to $Y,$ even if we impose no low$-$rank constraints. To guess values of $U,V$

that exactly multiply to $Y$ is like guessing values for the TA's horror$-$film fanhood that maximally explain the

TA's comment about The Shining : to do so fails to account for inherent plausibility and the presence of

competing explanations. END SIDENOTE $)$

Assume we start by fixing $V$ to initial values of $[4,2,1{]}^T.$ Find the optimal $21$ vector $U$ in this case.

$($Express your answer in terms of $).$

The first element of $U$ is:

The second element of $U$ is: