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)
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)
“I, who travel most often for my pleasure, do not direct myself so badly. If it looks ugly on the right, I take the left; if I find myself unfit to ride my horse, I stop.... Have I left something unseen behind me? I go back; it is still on my road. I trace no fixed line, either straight or crooked.”
—Michel de Montaigne (1533–1592)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“Time in his little cinema of the heart
Giving a première to Hate and Pain;
And Space urbanely keeping us apart.”
—Philip Larkin (1922–1986)