Question

If a subset B is a basis of a vector space V, then
(A). B generates V.
(B). B contains zero vector.
(C). B is linearly independent.
(D). B is the only basis of V.
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only.
• B. (A) and (C) only.
• C. (A), (B), (C) and (D).
• D. (C) and (D) only.

Hint:

Remember the two main properties of a basis: it must be a \textbf{linearly independent} set, and it must \textbf{span} the entire vector space. A good basis has just enough vectors to reach everywhere (span) but no redundant vectors (linearly independent).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A basis of a vector space V is a set of vectors that satisfies two fundamental properties. These properties ensure that any vector in V can be uniquely represented as a linear combination of the basis vectors.
Step 2: Detailed Explanation:
By definition, a subset B of a vector space V is a basis if and only if it satisfies the following two conditions:
B is linearly independent. This means that no vector in B can be written as a linear combination of the other vectors in B. This corresponds to statement (C).
B spans (or generates) V. This means that every vector in V can be written as a linear combination of the vectors in B. This corresponds to statement (A).
Let's analyze the other statements:
(B) B contains zero vector: This is false. Any set containing the zero vector is automatically linearly dependent, which violates the first condition of a basis.
(D) B is the only basis of V: This is false. Any non-trivial vector space has infinitely many bases. For example, in \(\mathbb{R}^2\), both \(\{(1,0), (0,1)\}\) and \(\{(1,1), (1,-1)\}\) are valid bases.
Step 3: Final Answer:
The two defining properties of a basis are that it is a linearly independent set and it generates (spans) the vector space. Therefore, statements (A) and (C) are correct.