Given the following data of sample size, N $=4,$ and $2$ features:

Feature Example $1$ Example $2$ Example $3$ Example $4$

X $48137$

Y $114514$

Use the principal component analysis $($PCA$)$ method to reduce the dimension from $2$ to $1.$ Subsequently, address the following steps:

$1.$ Compute the mean for each feature $($X$,$ and Y$).$

$2.$ Compute the covariance matrix S$.$

$3.$ Compute the eigen values, and eigen vectors for S$.$

$4.$ Determine the first principal component, and normalize its corresponding eigen vector.

$5.$ Derive and plot the new dataset