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:
“O, popular applause! what heart of man
Is proof against thy sweet, seducing charms?”
—William Cowper (1731–1800)
“The insatiable thirst for everything which lies beyond, and which life reveals, is the most living proof of our immortality.”
—Charles Baudelaire (1821–1867)
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)