Question

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

• A. \( \{ f \in V : f(1/2) € Q \} \) is a subspace of \( V \).
• B. \( \{ f \in V : f(1/2) = 1 \} \) is a subspace of \( V \).
• C. \( \{ f \in V : f(1/2) = f(1) \} \) is a subspace of \( V \).
• D. \( \{ f \in V : f'(1/2) = 1\} \) is a subspace of \( V \).

Answer 1

The Correct Option is C

To determine which of the given sets is a subspace of the real vector space \( V \) of polynomials in one variable with real coefficients and degree at most 6, we need to understand the subspace criteria. A subset \( S \) of a vector space \( V \) is a subspace if it satisfies three conditions:

1. Zero vector of \( V \) is in \( S \).
2. If \( u \) and \( v \) are elements of \( S \), then \( u + v \) is also in \( S \) (closure under addition).
3. If \( c \) is a scalar and \( u \) is an element of \( S \), then \( cu \) is also in \( S \) (closure under scalar multiplication).

Let's analyze each option:

1. \(\{ f \in V : f(1/2) \in \mathbb{Q} \}\)
   • The set includes polynomials for which the value attained at \( x = 1/2 \) is a rational number. Checking if this is a subspace, let \( f(1/2) = q_1 \) and \( g(1/2) = q_2 \) where \( q_1, q_2 \in \mathbb{Q} \). Then for any polynomials \( f, g \in V \), we should check if \( (f+g)(1/2) = f(1/2) + g(1/2) \) is also in \( \mathbb{Q} \). This is true owing to \( \mathbb{Q} \) being closed under addition. Now consider \( cf(1/2) = cq_1 \), which is also in \( \mathbb{Q} \) if \( c \in \mathbb{Q} \). However, if \( c \notin \mathbb{Q} \), \( cf(1/2) \) may not remain in \( \mathbb{Q} \). Therefore, this set is not a subspace.
2. \(\{ f \in V : f(1/2) = 1 \}\)
   • This set contains functions for which the value at \( x = 1/2 \) is exactly 1. The zero polynomial does not satisfy this since \( f(1/2) = 0 \). Thus, it fails the subspace criteria that require containing the zero vector.
3. \(\{ f \in V : f(1/2) = f(1) \}\)
   • Consider polynomials \( f, g \in V \) such that \( f(1/2) = f(1) \) and \( g(1/2) = g(1) \). For closure under addition: \((f+g)(1/2) = f(1/2) + g(1/2)\) and \((f+g)(1) = f(1) + g(1)\). Thus, \((f+g)(1/2) = (f+g)(1)\). For scalar multiplication: If \( cf(1/2) = cf(1) \) for any scalar \( c \), it holds true. The set satisfies all conditions of a subspace.
4. \(\{ f \in V : f'(1/2) = 1\}\)
   • This set consists of functions such that their derivative at \( x = 1/2 \) is 1. This does not include the zero polynomial, failing the subspace requirement.

The correct choice is: \(\{ f \in V : f(1/2) = f(1) \}\) is a subspace of \( V \).