Consider a set of data points $\{x_1,x_2,dots{x}_N\}$ where each $x_{i}in{R}^d,$ given to you after centering, i$.$e$.,\frac{1}{N}{\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i=0.$

Now suppose you want to project this data on a single unit vector given by $u$ by learning an appropriate $u.$ Show that, for learning $u,$ minimizing

the residual of the projections computed by the squared error between the projected data and the original data is equivalent to maximizing the

variance of the projections. Hint : the projection of an $x$ on a unit vector $u$ is given by $(x^{T}u)u.$ show work