Question

vector space of polynomials, let p be a set of polynomials from the vector space of polynomials over R. p={p1 (x)=2, p2 (x)=x, p3 (x)=10x²_5, p4 (x)=x³_3x determine wether the set p is linearly independent or not.

Answer 1

The set p is linearly independent in the vector space of polynomials over R.

Step-by-step explanation:

The set p is considered linearly independent if the only solution to the equation a1p1(x) + a2p2(x) + a3p3(x) + a4p4(x) = 0

is when a

1

= a

2

= a

3

= a

4

= 0.

Let's try to find the coefficients a1, a2, a3, and a4 that make the equation true:

1. 2a1 + xa2 + (10x²-5)a3 + (x³-3x)a4 = 0
2. 2a1 + xa2 + 10a3x² - 5a3 + (a4x³ - 3a4x) = 0
3. 2a1 + xa2 + 10a3x² - 5a3 + a4x³ - 3a4x = 0

We can see that we have a system of linear equations.
By equating the coefficients of like terms, we have the following equations:

1. 2a1 = 0
2. a2 = 0
3. 10a3 = 0
4. -5a3 = 0
5. a4 = 0
6. -3a4 = 0

These equations have a unique solution: a1 = a2 = a3 = a4 = 0. Therefore, the set p is linearly independent.