Relation To Bases
If the set is a frame of V, it spans V. Otherwise there would exist at least one non-zero which would be orthogonal to all . If we insert into the frame condition, we obtain
therefore, which is a violation of the initial assumptions on the lower frame bound.
If a set of vectors spans V, this is not a sufficient condition for calling the set a frame. As an example, consider and the infinite set given by
This set spans V but since we cannot choose . Consequently, the set is not a frame.
Read more about this topic: Frame Of A Vector Space
Famous quotes containing the words relation to, relation and/or bases:
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“Skepticism is unbelief in cause and effect. A man does not see, that, as he eats, so he thinks: as he deals, so he is, and so he appears; he does not see that his son is the son of his thoughts and of his actions; that fortunes are not exceptions but fruits; that relation and connection are not somewhere and sometimes, but everywhere and always; no miscellany, no exemption, no anomaly,—but method, and an even web; and what comes out, that was put in.”
—Ralph Waldo Emerson (1803–1882)
“The bases for historical knowledge are not empirical facts but written texts, even if these texts masquerade in the guise of wars or revolutions.”
—Paul Deman (1919–1983)