Determine whether thepolynomial 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 need to check if P can be expressed as a linear combination of a set of vectors that form a basis for P2.
In this case, P2 represents the set of all polynomials of degree at most 2. Any polynomial in P2 can be expressed in the form ax^2 + bx + c, where a, b, and c are real numbers.
To find the basis vectors e1, e2, and e3, we need to look for three particular polynomials in P2 that are linearly independent. This means that no one polynomial can be expressed as a linear combination of the other two.
A common choice for a basis for P2 is {1, x, x^2}, which represents the monomials of degree 0, 1, and 2, respectively. Therefore, we can take e1 = 1, e2 = x, and e3 = x^2.
Now, to determine if P can be expressed as a linear combination of e1, e2, and e3, we set up the equation:
4x^2 - x + 5 = a(1) + b(x) + c(x^2)
This equation represents the coefficients of the linear combination of e1, e2, and e3 that would equal P. To solve for a, b, and c, we equate the coefficients of like terms:
4 = 0 + 0 + c
-1 = 0 + b + 0
5 = a + 0 + 0
From the first equation, we get c = 4.
From the second equation, we get b = -1.
From the third equation, we get a = 5.
Now, substitute these values back into the equation:
4x^2 - x + 5 = 5(1) - 1(x) + 4(x^2)
= 5 - x + 4x^2
We can see that the polynomial P can indeed be expressed as a linear combination of e1, e2, and e3, since it matches the right-hand side of the equation.
Therefore, the polynomial P = 4x^2 - x + 5 spans P2.