4) Consider the field \( \mathbb{R} \) of real numbers. Let \( x \) denote an indeterminate, and consider the
vector space \( V \) over \( \mathbb{R} \) consisting of polynomials in \( x \) that have coeficients in \( \mathbb{R} \) and degree
at most \( 3. \) Note that \( \{1, x, x^2, x^3\} \) is a basis for \( V. \) Consider the map \( \varphi : V \rightarrow \mathbb{R} \) that sends
\( f \mapsto f(2) \) for all \( f \in V. \; \) (i) Prove that \( \varphi \) is in the dual space \( V^*. \; \) (ii) Express \( \varphi \) as a linear
combination of the dual basis.
Proof. Let \( f(x), g(x) \in V = \{a_0 + a_1 x + a_2 x^2 + a_3 x^3 \; | \; a_i \in \mathbb{R}\} \). Then \( \varphi (f(x) + g(x)) = \)
\( \varphi ((f + g)(x)) = (f + g)(2) = f(2) + g(2) = \varphi (f(x)) + \varphi (g(x)). \) And \( \varphi( \alpha f(x)) \) = \( (\alpha f)(2) = \)
\( \alpha f(2) = \alpha \varphi(f(x)). \) Therefore \( \varphi \) is a linear transformation; ie., \( \varphi \in V^*. \)
For (ii), let \( p(x) = {a_0 + a_1 x + a_2 x^2 + a_3 x^3} \) and put \( v_i = x^i, i = 0, 1, 2, 3. \) Then \( v_0^*(p(x)) = \)
\( v_0^*(a_0 + a_1 x + a_2 x^2 + a_3 x^3) = a_0, \) considering that \( v_0^* \) is \( 1 \) on \( 1 \) and \( 0 \) on other powers
of \( x. \) In the same vein, \( v_i^*(p(x)) = a_i \) for all \( i. \)
That established, \( \varphi (p(x)) = \varphi(a_0 + a_1 x + a_2 x^2 + a_3 x^3) = a_0 + 2 a_1 + 2^2 a_2 + 2^3 a_3 = \)
\( v_0^*(p(x)) + 2 \cdot v_1^*(p(x)) + 2^2 \cdot v_2^*(p(x)) + 2^3 \cdot v_3^*(p(x)). \) QED.
Note: This is similar to part of Ex. §11.3,#2 in Dummit & Foote.