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 the, relation to, relation, left, null and/or space:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“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 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 have friends, you know, one need only be good-natured; but when a man has no enemy left there must be something mean about him.”
—Oscar Wilde (1854–1900)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“Sir Walter Raleigh might well be studied, if only for the excellence of his style, for he is remarkable in the midst of so many masters. There is a natural emphasis in his style, like a man’s tread, and a breathing space between the sentences, which the best of modern writing does not furnish. His chapters are like English parks, or say rather like a Western forest, where the larger growth keeps down the underwood, and one may ride on horseback through the openings.”
—Henry David Thoreau (1817–1862)