Basis Properties for Vector Space V

This question requires understanding the fundamental definition of a basis for a vector space and applying it to evaluate given properties.

Vector Space Basis Definition

In linear algebra, a subset $B$ of a vector space $V$ is formally defined as a basis for $V$ if it satisfies two essential criteria:

• Spanning Property: The set $B$ must generate the entire vector space $V$. This means that any vector in $V$ can be expressed as a finite linear combination of the vectors present in $B$.
• Linear Independence Property: The set $B$ must be linearly independent. This condition ensures that no vector in $B$ can be represented as a linear combination of the other vectors in $B$. Mathematically, the only solution to the equation $\sum_{i=1}^{n} c_i b_i = \vec{0}$, where $b_i \in B$ and $c_i$ are scalars, is $c_i = 0$ for all $i$.

Basis B Statements Analysis

• Statement (A): B generates V.

  This statement is true. It directly reflects the spanning property, which is one of the two defining conditions for a set to be a basis of a vector space.

• Statement (B): B contains zero vector.

  This statement is false. A set that includes the zero vector ($\vec{0}$) cannot be linearly independent. For any set containing $\vec{0}$, we can always form a linear combination resulting in the zero vector using a non-zero coefficient (e.g., $1 \cdot \vec{0}$ = \vec{0}$). Since a basis must be linearly independent, it cannot contain the zero vector.

• Statement (C): B is linearly independent.

  This statement is true. Linear independence is the second fundamental condition required for a set to qualify as a basis for a vector space.

• Statement (D): B is the only basis of V.

  This statement is false. A vector space can, and often does, have multiple distinct bases. While all bases for a specific finite-dimensional vector space contain the same number of vectors (which defines the dimension of the space), the actual vectors forming the basis can differ. For example, in the vector space $\mathbb{R}^2$, both $\{(1, 0), (0, 1)\}$ and $\{(1, 1), (1, -1)\}$ are valid bases.

Basis Combination Choice

Based on the analysis, the properties that must hold true for a subset $B$ to be a basis of a vector space $V$ are that $B$ generates $V$ (Statement A) and $B$ is linearly independent (Statement C).

Statements (B) and (D) are incorrect properties of a basis.

Therefore, the correct option must include only statements (A) and (C).