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:
“The adolescent does not develop her identity and individuality by moving outside her family. She is not triggered by some magic unconscious dynamic whereby she rejects her family in favour of her peers or of a larger society.... She continues to develop in relation to her parents. Her mother continues to have more influence over her than either her father or her friends.”
—Terri Apter (20th century)
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“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)