Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $P_0(t) = 1, P_1(t) = t, P_2(t) = t^2, P_3(t) = t^3$, then $\{P_0, P_1, P_2, P_3\}$ is linearly independent. Reason R : $C_0 P_0 + C_1 P_1 + C_2 P_2 + C_3 P_3 = 0 \implies C_0 = 0, C_1 = 0, C_2 = 0, C_3 = 0$.
In the light of the above statements, choose the correct answer from the options given below

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

Standard monomial sets $\{1, t, t^2, \dots, t^n\}$ are always linearly independent in any polynomial space $\mathcal{P}_n(\mathbb{R})$.

Answer 1

The Correct Option is A

Step 1: Concept
By definition, a set of functions or vectors $\{v_1, v_2, \dots, v_n\}$ in a vector space $V$ is linearly independent if the only linear combination yielding the zero vector is the trivial one ($c_1 = c_2 = \dots = c_n = 0$).

Step 2: Key Formulas and Approach
Test the linear dependence relation $C_0 P_0(t) + C_1 P_1(t) + C_2 P_2(t) + C_3 P_3(t) = 0$ for all $t \in \mathbb{R}$.

Step 3: Step-by-step Explanation

• Evaluating Assertion A: The set of polynomials $\{1, t, t^2, t^3\}$ forms the standard monomial basis for $P_3(\mathbb{R})$, the space of polynomials of degree at most 3. No polynomial in this set can be expressed as a linear combination of the others. Thus, Assertion A is true.
• Evaluating Reason R: Set $C_0(1) + C_1(t) + C_2(t^2) + C_3(t^3) = 0$ for all $t$. By the Fundamental Theorem of Algebra, a non-zero polynomial of degree $n$ has at most $n$ roots. Since this polynomial equals $0$ for infinitely many values of $t$, all coefficients must be identically zero: \[ C_0 = 0, \quad C_1 = 0, \quad C_2 = 0, \quad C_3 = 0 \] Hence, Reason R is true.
• Checking Explanation: Reason R states the exact mathematical definition of linear independence applied to the set $\{P_0, P_1, P_2, P_3\}$, proving Assertion A directly. Thus, R is the correct explanation of A.

Step 4: Final Answer
Both A and R are true and R is the correct explanation of A. Thus, Option (A) is correct.