Positional encoding. Consider the sentence "I love Machine Learning". Here, we treat each word as a token, where "I"

has an index value of $1,$ "love" has an index value of $2,$ and so on$.$ To construct the positional encoding matrix $W_p$ for this

sentence, we will fix the embedding$/$concept space to be $4-$dimensional such that $W_{p}in{R}^{44}$ and use a modified version

of the positional encoding function proposed in the original paper "Attention is All you Need" by Vaswani et al$.,$ namely,

for any $i,$jin$\{1,$dots,$4\}$

$(W_p{)}_{ij}=\{\begin{array}{ccc}cos(\frac{i}{10000\frac{|\frac{j}{2}|}{2}}),& ifjis\text{o}\text{d}\text{d}\text{,}& \\ sin(\frac{i}{{10000}^{\frac{|\frac{j}{2}|}{2}}}),& ifjis\text{e}\text{v}\text{e}\text{n}\text{,}& \end{array}$

where $|$~$z$~$|$ is the smallest integer which is greater than or equal to $z.$ Using this encoding function, write down the positional

encoding matrix.