alleryStart learning →A BASIS of a vector space $V$ is a set that is
Aorthogonal only
Bthe set of all eigenvectors
Cinfinite
Dlinearly independent AND spans $V$
Answer & Solution
Correct answer: D. linearly independent AND spans $V$
1. A BASIS is a SET that is (a) LINEARLY INDEPENDENT and (b) SPANS the whole space $V$.
2. Every vector $v \in V$ has a UNIQUE expansion as a linear combination of basis vectors.
3. The number of vectors in a basis is the DIMENSION of $V$ — independent of which basis you choose.
4. Option A is for orthogonal bases (special case). Options C, D are not necessary properties.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §2.3 (Basis)._
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