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Dimension theorem for vector spaces
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the dimension theorem for vector spaces states that all <a href="/facts/Basis_(linear_algebra)/89IPoN6c">bases</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a>), and defines the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of the vector space. </p><p>Formally, the dimension theorem for vector spaces states that: </p> Given a vector space <i>V</i>, any two bases have the same <a href="/facts/Cardinality/m6VPojwT">cardinality</a>. <p>As a basis is a <a href="/facts/Generating_set/Jo1eBRbD">generating set</a> that is <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a>, the dimension theorem is a consequence of the following <a href="/facts/Theorem/f2Pe1FMD">theorem</a>, which is also useful: </p> In a vector space <i>V</i>, if G is a generating set, and I is a linearly independent set, then the cardinality of I is not larger than the cardinality of G. <p>In particular if <i>V</i> is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a>, then all its bases are finite and have the same number of elements. </p><p>While the <a href="/facts/Mathematical_proof/7ECDrU80">proof</a> of the existence of a basis for any vector space in the general case requires <a href="/facts/Zorn%2527s_lemma/ssCublFT">Zorn's lemma</a> and is in fact equivalent to the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, the uniqueness of the cardinality of the basis requires only the <a href="/facts/Ultrafilter_lemma/KUpERdqN">ultrafilter lemma</a>, which is strictly weaker (the proof given below, however, assumes <a href="/facts/Trichotomy_(mathematics)/o3mHmLka">trichotomy</a>, i.e., that all <a href="/facts/Cardinal_number/ccoKwU4R">cardinal numbers</a> are comparable, a statement which is also equivalent to the axiom of choice). The theorem can be generalized to arbitrary <a href="/facts/Module_(mathematics)/jP0LIhFL"><i>R</i>-modules</a> for rings <i>R</i> having <a href="/facts/Invariant_basis_number/gv0n7JEa">invariant basis number</a>. </p><p>In the finitely generated case the proof uses only elementary arguments of <a href="/facts/Algebra/nMdqczjo">algebra</a>, and does not require the axiom of choice nor its weaker variants. </p>