$1.$ 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.

$1.$ 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.

$2.$ 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.

$3.$ 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?