Proof
Assume that { ai: i ∈ I } and { bj: j ∈ J } are both bases, with the cardinality of I bigger than the cardinality of J. From this assumption we will derive a contradiction.
Read more about this topic: Dimension Theorem For Vector Spaces
Famous quotes containing the word proof:
“The proof of a poet is that his country absorbs him as affectionately as he has absorbed it.”
—Walt Whitman (1819–1892)
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)