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:
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“... a worker was seldom so much annoyed by what he got as by what he got in relation to his fellow workers.”
—Mary Barnett Gilson (1877–?)
“Historically and politically, the petit-bourgeois is the key to the century.... The bourgeois and proletariat classes have become abstractions: the petite-bourgeoisie, in contrast, is everywhere, you can see it everywhere, even in the areas of the bourgeois and the proletariat, what’s left of them.”
—Roland Barthes (1915–1980)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“What a phenomenon it has been—science fiction, space fiction—exploding out of nowhere, unexpectedly of course, as always happens when the human mind is being forced to expand; this time starwards, galaxy-wise, and who knows where next.”
—Doris Lessing (b. 1919)