Figure $1$: Dataset $1$

Figure $2$: Dataset $2$

$($b$)$ Is it possible to have a data set in $R^2$ that is linearly separable by a halfspace in $R^2$ but

is not linearly separable after projecting onto either of the two principal components?

If so$,$ give a simple example along the lines of the above data sets. If not, explain in

$1-2$ sentences why it is not possible.

$($a$)$ In each of the following plots, a training set of datia points $x$ in $R{L}^2$ labeled cither $+$

or $-$ is given, where the original features are the coordinates $(x,y).$ You can assume

that the data is origin$-$centered. For each of the two training sets below, answer the

$[$ollowing questions:

$($i$)$ Draw a simple simple recrention of cand of the two datasets below $($no need for

exact precision$)$ and draw all the principal components $($cyeball th$).$

$($ii$)$ For each dataset, can we correctly classify the labels by nsing halispace after

projecting outo one of the primcipal components? IC so$,$ which principal component

should we projoct onta? If not, cxplain in $1-2$ gentenoes why it is not possible.

Figure $1$: Dataset $1$