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:
“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)
“Skepticism is unbelief in cause and effect. A man does not see, that, as he eats, so he thinks: as he deals, so he is, and so he appears; he does not see that his son is the son of his thoughts and of his actions; that fortunes are not exceptions but fruits; that relation and connection are not somewhere and sometimes, but everywhere and always; no miscellany, no exemption, no anomaly,—but method, and an even web; and what comes out, that was put in.”
—Ralph Waldo Emerson (1803–1882)
“I never saw so sweet a face
As that I stood before:
My heart has left its dwelling-place
And can return no more.”
—John Clare (1793–1864)
“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)