Question 4.1.10: Show that the polynomials of degree ≤2, of the form f (x) = ......

Linear Algebra with Applications [1016048]

Show that the polynomials of degree ≤2, of the form $f (x) = a + bx + cx^2$ , are a subspace W of the space F(ℝ, ℝ) of all functions from ℝ to ℝ.

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a. W contains the neutral element of F(ℝ, ℝ), the zero function f (x) = 0. Indeed, we can write $f (x) = 0 + 0x + 0x^2$ .

b. W is closed under addition: If two polynomials $f (x) = a + bx + cx^2$ and $g(x) = p + qx + r x^2$ are in W, then their sum $f (x) + g(x) = (a + p) + (b + q)x + (c + r)x^2$ is in W as well, since f (x) + g(x) is a polynomial of degree ≤2.

c. W is closed under scalar multiplication: If $f (x) = a + bx + cx^2$ is a polynomial in W and k is a constant, then $k f (x) = ka + (kb)x + (kc)x^2$ is in W as well.

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