Generator of Vector Space is Basis iff Cardinality equals Dimension
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Theorem
Let $E$ be a vector space of $n$ dimensions.
Let $G$ be a generator for $E$:
- $G$ is a basis for $E$ if and only if $\card G = n$.
Proof
Necessary Condition
Let $G$ be a basis for $E$.
From Cardinality of Basis of Vector Space, $\card G = n$.
$\Box$
Sufficient Condition
Let $\card G = n$.
From Sufficient Conditions for Basis of Finite Dimensional Vector Space, $G$ is a basis for $E$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.14$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 34$. Dimension: Theorem $67 \ \text{(iv)}$