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:
“Hesitation increases in relation to risk in equal proportion to age.”
—Ernest Hemingway (1899–1961)
“Our sympathy is cold to the relation of distant misery.”
—Edward Gibbon (1737–1794)
“I was glad to have got out of the towns, where I am wont to feel unspeakably mean and disgraced,—to have left behind me for a season the bar-rooms of Massachusetts, where the full-grown are not weaned from savage and filthy habits,—still sucking a cigar. My spirits rose in proportion to the outward dreariness. The towns needed to be ventilated. The gods would be pleased to see some pure flames from their altars. They are not to be appeased with cigar-smoke.”
—Henry David Thoreau (1817–1862)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“The peculiarity of sculpture is that it creates a three-dimensional object in space. Painting may strive to give on a two-dimensional plane, the illusion of space, but it is space itself as a perceived quantity that becomes the peculiar concern of the sculptor. We may say that for the painter space is a luxury; for the sculptor it is a necessity.”
—Sir Herbert Read (1893–1968)