Determine whether the polynomial P=4x^2-x+5 spans P2.
Soln:
4x^2-x+5 = a(e1)+b(e2)+c(e3)
...but I do not know how to obtain e1, e2 and e3. That's the main problem.
To determine whether the polynomial P = 4x^2 - x + 5 spans P2, we first need to understand what P2 represents. P2 denotes the set of all polynomials of degree 2 or less.
To find the basis elements e1, e2, and e3 of P2, we need to look for polynomials that have a degree of 2 or less and are linearly independent. Linear independence means that no polynomial in the set can be represented as a linear combination of the other polynomials.
A common approach to finding the basis elements of P2 involves considering monomials of the form ax^2, bx, and c, where a, b, and c are constants. These monomials form the standard basis of P2.
So, in our case, e1 = x^2, e2 = x, and e3 = 1 form a potential basis for P2.
Now, let's substitute these basis elements into the equation for P and solve for the coefficients a, b, and c.
P = 4x^2 - x + 5
= a(x^2) + b(x) + c(1)
By comparing corresponding coefficients, we get the following system of equations:
4 = a (coefficient of x^2)
-1 = b (coefficient of x)
5 = c (constant coefficient)
Solving this system of equations, we find that a = 4, b = -1, and c = 5.
Therefore, we can express P as a linear combination of the basis elements e1, e2, and e3:
P = 4(x^2) - 1(x) + 5(1)
= 4x^2 - x + 5
Since P can be expressed as a linear combination of the basis elements e1, e2, and e3 with coefficients a = 4, b = -1, and c = 5, we can conclude that the polynomial P spans P2.