Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Set $\{(1,0,0), (0,1,0), (0,0,1)\}$ forms a basis of $V_3(\mathbb{R})$. Reason R : Let $V$ be a vector space over a field $F$ and $S$ be a non-empty subset of $V$, then $S$ is a basis of $V$ if
(i) $S$ is linearly dependent
(ii) $\text{span}(S) = V$. 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:

Read assertion-reason questions carefully! A single incorrect word like "dependent" instead of "independent" makes a mathematical reason completely false.

Answer 1

The Correct Option is C

Step 1: Concept
A non-empty subset $S$ of a vector space $V$ is defined as a basis of $V$ if and only if two conditions are met: 1. $S$ is linearly independent. 2. $S$ spans $V$, i.e., $\text{span}(S) = V$.

Step 2: Key Formulas and Approach
We evaluate Assertion A and Reason R independently based on the formal definition of a vector space basis.

Step 3: Step-by-step Explanation

• Evaluating Assertion A:
The set $S = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ is the standard basis of $\mathbb{R}^3 = V_3(\mathbb{R})$.
- It is linearly independent because $c_1(1,0,0) + c_2(0,1,0) + c_3(0,0,1) = (0,0,0) \implies c_1 = c_2 = c_3 = 0$.
- It spans $V_3(\mathbb{R})$ because any vector $(x, y, z) = x e_1 + y e_2 + z e_3$.
Hence, Assertion A is true.

• Evaluating Reason R:
Reason R states that $S$ is a basis if "(i) $S$ is linearly dependent".
This is incorrect! A basis requires $S$ to be linearly independent. Linear dependence prevents a set from being a basis.
Hence, Reason R is false.

Step 4: Final Answer
Assertion A is true, but Reason R is false. Thus, Option (C) is correct.