Question

Let P₁₁(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 11, together with the zero polynomial. Let
E = {s₀(x), s₁(𝑥), … , s₁₁(x)}, F = {r0 (x), r₁ (x), … , r₁₁(x)}
be subsets of P₁₁(x) having 12 elements each and satisfying
s₀ (3) = s₁ (3) = ⋯ = s₁₁(3) = 0 , r₀(4) = r₁(4) = ⋯ = r₁₁(4) = 1.
Then, which one of the following is TRUE ?

• A. Any such E is not necessarily linearly dependent and any such F is not necessarily linearly dependent
• B. Any such E is necessarily linearly dependent but any such F is not necessarily linearly dependent
• C. Any such E is not necessarily linearly dependent but any such F is necessarily linearly dependent
• D. Any such E is necessarily linearly dependent and any such F is necessarily linearly dependent

Answer 1

The Correct Option is B

- Statement \( E \): The set \( E \) consists of 12 polynomials in \( P_{11}(x) \), which are polynomials of degree at most 11. The space \( P_{11}(x) \) has dimension 12, and therefore any set of 12 polynomials in this space must be linearly dependent (since the maximum number of linearly independent vectors in \( P_{11}(x) \) is 12). Thus, \( E \) is necessarily linearly dependent. - Statement \( F \): The set \( F \) consists of 12 polynomials such that \( r_0(4) = r_1(4) = \dots = r_{11}(4) = 1 \), which imposes a condition at \( x = 4 \). However, the fact that all elements of \( F \) satisfy this condition does not guarantee linear dependence. The polynomials can still be linearly independent, so \( F \) is not necessarily linearly dependent. Thus, the correct answer is (B): Any such \( E \) is necessarily linearly dependent but any such \( F \) is not necessarily linearly dependent.