Question

Which of the following statement is always true

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

Hint:

In vector space theory, the boundary cases are defined as:
- The empty set \( \emptyset \) is linearly independent (its span is \( \{ \mathbf{0} \} \)).
- Any set containing the zero vector \( \mathbf{0} \) is linearly dependent.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In linear algebra, a set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k \} \) is linearly independent if the vector equation:
\[ c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k = \mathbf{0} \]
can only be satisfied by the trivial solution, \( c_1 = c_2 = \dots = c_k = 0 \).
If there is a non-trivial solution (where at least one coefficient is non-zero), the set is linearly dependent.

Step 2: Detailed Explanation:
Let us evaluate each of the statements:
- Option A: "Every single non-zero vector is linearly dependent."
Let \( \{ \mathbf{v} \} \) be a set containing a single non-zero vector.
The equation is \( c \cdot \mathbf{v} = \mathbf{0} \).
Since \( \mathbf{v} \neq \mathbf{0} \), the only solution is \( c = 0 \).
Therefore, any single non-zero vector is linearly independent, making Option A false.
- Option B and C: "Every empty set is linearly independent/dependent."
By definition, a set is linearly dependent if there exists a non-trivial linear combination of its elements that equals the zero vector.
An empty set \( \emptyset \) contains no vectors, so it is impossible to form any linear combination, let alone a non-trivial one.
Therefore, because the condition for linear dependence cannot be met, the empty set \( \emptyset \) is defined as linearly independent. Thus, Option C is always true.
- Option D: "Every non-empty set is linearly independent."
This is false; for example, the set \( \{ (1, 0), (2, 0) \} \) is linearly dependent because \( -2(1,0) + 1(2,0) = (0,0) \).

Step 3: Final Answer:
Every empty set is linearly independent.