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:
“Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.”
—Ralph Waldo Emerson (1803–1882)
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“Light is meaningful only in relation to darkness, and truth presupposes error. It is these mingled opposites which people our life, which make it pungent, intoxicating. We only exist in terms of this conflict, in the zone where black and white clash.”
—Louis Aragon (1897–1982)
“There seems to be a common strain of miserliness in the American people when it comes to throwing away toothpaste tubes which have a little left in the bottom.”
—Robert Benchley (1889–1945)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“Mere human beings can’t afford to be fanatical about anything.... Not even about justice or loyalty. The fanatic for justice ends by murdering a million helpless people to clear a space for his law-courts. If we are to survive on this planet, there must be compromises.”
—Storm Jameson (1891–1986)