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: Basis Fundamentals:
A basis in vector space V is a set of vectors. This set must fulfill two key properties to ensure any vector in V can be uniquely expressed as a linear combination of its basis vectors.
Step 2: Basis Properties Explained:
A subset B of vector space V constitutes a basis if and only if:
B is linearly independent. No vector in B can be created from a linear combination of other vectors within B. This corresponds to statement (C).
B spans V. Every vector in V can be written as a linear combination of vectors from B. This corresponds to statement (A).
Analyzing other statements:
(B) B contains the zero vector: This is incorrect. A set including the zero vector is always linearly dependent, breaking the first basis condition.
(D) B is the unique basis of V: This is also incorrect. Most vector spaces have infinitely many bases. For instance, in \(\mathbb{R}^2\), both \(\{(1,0), (0,1)\}\) and \(\{(1,1), (1,-1)\}\) are valid bases.
Step 3: Conclusion:
A basis has two essential properties: linear independence and spanning the vector space. Therefore, statements (A) and (C) are correct.