We want to approximate the original samples with three features using principal component analysis

$($PCA$).$ Assume that the scatter matrix obtained from the samples is

S $=$

$$

$$

$$

$7\mathrm{.}52\mathrm{.}50$

$2\mathrm{.}57\mathrm{.}50$

$002$

$$

$$

$.$

$($a$)$ Determine the first principal component w$.(13$ pts$)$

$($b$)$ Find the approximation of a sample x $=[1\mathrm{.}21\mathrm{.}7-1\mathrm{.}3]$T using the first principal component from