If they have the same basis, then surely they span the same vectors and so are equal?
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5Yes, that's right. – David Mitra Jan 31 '16 at 17:12
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Does this answer your question? Proof that the all bases of a subspace V consist of the same number of vectors? – Jul 27 '20 at 21:28
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@AnalysisJD not a duplicate, please read the question. – KReiser Jul 28 '20 at 00:29
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Suppose the basis of two spaces $S_1 \text{ and } S_2$ is $b = \{v_1,...,v_n\}$
If $a\in S_1$ then $a = a_1v_1+...+a_nv_n \in S_2$ similarly if $a'\in S_2$ then $ a' = a_1'v_1+...+a_n'v_n\in S_1$
What can you conclude?
fosho
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