Question

Which of the following statement is true?

• A. Every empty set is linearly dependent
• B. Every non empty set is linearly independent
• C. Every non empty set is linearly dependent
• D. Single non-zero set is linearly independent

Hint:

A set consisting of a single vector is:
- Linearly Independent if and only if the vector is non-zero (\(v \neq 0\)).
- Linearly Dependent if and only if the vector is the zero vector (\(v = 0\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In linear algebra, a set of vectors \(\{v_1, v_2, \dots, v_k\}\) is linearly independent if the vector equation:
\[ c_1 v_1 + c_2 v_2 + \dots + c_k v_k = 0 \] has only the trivial solution \(c_1 = c_2 = \dots = c_k = 0\).
If there exists a non-trivial solution where at least one coefficient is non-zero, the set is linearly dependent.

Step 3: Detailed Explanation:
Let us evaluate each of the given statements:
- Option (A): Every empty set is linearly dependent
By definition, the empty set \(\emptyset\) is considered linearly independent because there are no vectors in it to form a non-trivial linear combination that sums to zero. Thus, Option (A) is false.
- Option (B) and (C): Every non-empty set is linearly independent dependent
These are false generalization statements.
For example, in \(\mathbb{R}^2\), the non-empty set \(\{(1,0), (2,0)\}\) is linearly dependent, while \(\{(1,0), (0,1)\}\) is linearly independent.
- Option (D): Single non-zero set is linearly independent
Let us consider a set containing a single non-zero vector:
\[ S = \{ v \} \quad \text{where } v \neq 0 \]
To test for linear independence, we write the vector equation:
\[ c \cdot v = 0 \]
Since \(v\) is a non-zero vector (\(v \neq 0\)), the only way this product can equal the zero vector is if the scalar coefficient is zero:
\[ c = 0 \]
Because the equation has only the trivial solution \(c = 0\), the set \(S\) is linearly independent.
This statement is always true.

Step 4: Final Answer:
Therefore, the correct option is (D).