A data analysis team works with a dataset comprising three features:

Price, quality, and popularity $of$ various products. Each product can $be$

represented $as$ a vector $in$ a three$-$dimensional vector space, where each dimen$-$

sion corresponds $to$ one $of$ the three features.

Given the following three product vectors:

$o$ Product $1$: $P_1=(20,8,5)$

$o$ Product $2$: $P_2=(15,7,4)$

$o$ Product $3$: $P_3=(30,10,7)$

$a.$ Determine $if$ these vectors are linearly independent.

The team wants $to$ analyze a new product represented $by$ the vector $P_4=$

$(25,9,6).$

$b.If$ possible, Express $P_{4}as$ a linear combination $of$ the other three product

vectors.

$If{P}_4$ can $be$ expressed $as$ a linear combination, discuss what this implies

about the new product $in$ relation $to$ the original products.

Follow$-up$:

How could this analysis inform the team's strategy for marketing $or$ pricing the

new product?