Let \(V\) be the (real) vector space of all polynomial functions from \(\RR\) into \(\RR\) of degree 2 or less, i.e., the space of all functions \(f\) of the form \[ f(x) = c_0 + c_1 x + c_2 x^2. \] Let \(t\) be a fixed real number and define \[ g_1(x) = 1, \quad g_2(x) = x + t, \quad g_3(x) = (x + t)^2. \] Prove that \(\mathfrak{B} = \{g_1, g_2, g_3\}\) is a basis for \(V\). If \[ f(x) = c_0 + c_1 x + c_2 x^2, \] what are the coordinates of \(f\) in this ordered basis \(\mathfrak{B}\)?
Solution
The functions \[ g_1(x)=1, \qquad g_2(x)=x+t, \qquad g_3(x)=(x+t)^2 \] span \(V\): if \[ f(x)=c_0+c_1x+c_2x^2, \] then \[ f(x)=a+b(x+t)+c(x+t)^2 \] with \[ c=c_2, \qquad b=c_1-2tc_2, \qquad a=c_0-tc_1+t^2c_2. \] Indeed, expanding the right side gives \[ (a+bt+ct^2)+(b+2ct)x+cx^2, \] whose coefficients match \(f\) for exactly these values of \(a,b,c\).
The representation is unique because the coefficient comparison determines \(a,b,c\) uniquely. Therefore \(\{g_1,g_2,g_3\}\) is a basis, and \[ [f]_{\mathfrak B} = \begin{bmatrix} c_0-tc_1+t^2c_2\\ c_1-2tc_2\\ c_2 \end{bmatrix}. \]