Given the data points in $2$D space: $(1,1),(2,2)$ and $(3,3),$

a$)$ What is the first principle component?

Follow the steps in the lecture slides on dimensionality reduction $($slide $50).$

$($subtract the mean, calculate the covariance matrix, find the eigenvectors etc.$)$

b$)$ Project the data onto $1$D space given by the first principle component, what is the

variance of the projected data?

c$)$ For the projected data in $($b$),$ now if we represent them in the original $2$D space, what

is the reconstruction error?