Relation To The Left Null Space
The left null space of A is the set of all vectors x such that xTA = 0T. It is the same as the null space of the transpose of A. The left null space is the orthogonal complement to the column space of A.
This can be seen by writing the product of the matrix and the vector x in terms of the dot product of vectors:
where c1, ..., cn are the column vectors of A. Thus x = 0 if and only if x is orthogonal (perpendicular) to each of the column vectors of A.
It follows that the null space of is the orthogonal complement to the column space of A.
For a matrix A, the column space, row space, null space, and left null space are sometimes referred to as the four fundamental subspaces.
Read more about this topic: Column Space
Famous quotes containing the words relation to, relation, left, null and/or space:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“Art should exhilarate, and throw down the walls of circumstance on every side, awakening in the beholder the same sense of universal relation and power which the work evinced in the artist, and its highest effect is to make new artists.”
—Ralph Waldo Emerson (1803–1882)
“He was last seen flying to New York.
He was handing out cards which read:
“He wears a question in his left eye.
He dislikes the police but will associate with them.
He will demand something not on the menu.
He is invisible to the eyes of beauty and culture....””
—John Ashbery (b. 1927)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“To play is nothing but the imitative substitution of a pleasurable, superfluous and voluntary action for a serious, necessary, imperative and difficult one. At the cradle of play as well as of artistic activity there stood leisure, tedium entailed by increased spiritual mobility, a horror vacui, the need of letting forms no longer imprisoned move freely, of filling empty time with sequences of notes, empty space with sequences of form.”
—Max J. Friedländer (1867–1958)