Determine whether the set {x2 + 2x − 3, 4x2 + x − 1, 5x2 + 4x − 4} is a basis for p2 by solving the equation a(x2 + 2x − 3) + b(4x2 + x − 1) + c(5x2 + 4x − 4) = 0 for a, b, and c.

The set is a basis for P2

Step-by-step explanation:

Remember, a set B is a basis for a vector space V if B is a linear independent set and gen(B)=V. Also, remember that the polynomials are linearly independent if the unique scalars such that

are

Expanding the above equation we have that

Then we need values of a, b and c such that

So, we have a homogeneous system with augmented matrix

Now, we apply row operations for solve the system

1. To the second row of A we subtract the first row twice. And to the third row we add the first row three times. We get the matrix:

2. We multiply the third row of the previous matrix by 1/11

3. To the third row of the previous matrix we add 7 times the second row.

Since the rank of the matrix is 3 then the system has unique solution. But the system is homogeneous, therefore the solution is

This show that the polynomials are linear independent and therefore the set is a basis for P2

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step-by-step explanation:

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