Question

Prove that the following two sets are the same. S1 = {a + bx : a, b ∈ R} = all polynomials which can expressed as a linear combination of 1 and x; S2 = {ax + b(2 + x) : a, b ∈ R} = all polynomials which can expressed as a linear combination of x and 2 + x.

Answer 1

slope-intercept formula y=mx+b

Answer 2

Answer with Step-by-step explanation:

We are given that twos sets

`S_1`={a+bx:
`a,b\in R`}=All polynomials which can expressed as a linear combination of 1 and x.

`S_2`={ax+b(2+x):
`a,b\in R`}=All polynomials which can be expressed as a linear combination of x and 2+x.

We have to prove that given two sets are same.

`S_2`={ax+2b+bx}={(a+b)x+2b}={cx+d}

`S_2`={cx+d}=All polynomials which can be expressed as a linear combination of 1 and x.

Because a+b=c=Constant

2b= Constant=d

Hence, the two sets are same .