Problem $1$: Sparse Vector Representation

In this problem, you will get practice with converting between sparse and dense vector representations.

You will start by converting dense vectors into a sparse representation. Consider the following vectors:

$1.[1,0,0,1,1,0,0,0]$

$2.[0,0,4,0,7,0,0,0,2,0]$

$3.[0,0,1,0,0,0,0,0,1,0]$

$4.[4,1,8]$

$5.[0,0,0,0,0,0]$

Create a markdown cell. Within this cell, create a numbered list showing sparse vector representations for each

of the vectors shown above. Place a "backtick" character before and after each list you provide as part of your

answer. This will format the answers in a mono$-$spaced font. For example, the markdown for a hypothetical

answer might be typed as follows: $`[6,[3,5],[2,1]]`$

Next, you will convert sparse vectors into a dense representation. Consider the following vectors:

$1.[8,[2,5,7],[1,3,9]]$

$2.[4,[],[]]$

$3.[4,[1,2,3],[2,3,1]]$

$4.[8,[1,5],[1,1]]$

$5.[6,[4],[2]]$

Add a new markdown cell. Create a numbered list showing dense vector representations for each of the sparse

vectors shown above. Place a "backtick" character before and after each list you provide as part of your answer.