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:
“If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a Declaration &c. which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
—Thomas Jefferson (17431826)
“The moment a man begins to talk about technique thats proof that he is fresh out of ideas.”
—Raymond Chandler (18881959)
“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 twoa proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)