A vector space $V$ has dimension 5. The maximum number of LINEARLY INDEPENDENT vectors that can be chosen from $V$ is
A5
Binfinity
C4
D10
Answer & Solution
Correct answer: A. 5
1. The DIMENSION of a vector space is the SIZE of any basis.
2. KEY THEOREM: any set of more than $\dim V$ vectors in $V$ is LINEARLY DEPENDENT.
3. So the MAXIMUM number of linearly independent vectors in $V$ is exactly $\dim V = 5$.
4. Any 5 linearly independent vectors form a BASIS of $V$.
5. Other options either underestimate or contradict the dimension bound.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §2.3 (Bases and dimension theorems)._
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