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:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“Hesitation increases in relation to risk in equal proportion to age.”
—Ernest Hemingway (1899–1961)
“To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
—Henry James (1843–1916)
“If we were left solely to the wordy wit of legislators in Congress for our guidance, uncorrected by the seasonable experience and the effectual complaints of the people, America would not long retain her rank among the nations.”
—Henry David Thoreau (1817–1862)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.”
—Ralph Waldo Emerson (1803–1882)