The left null space of a matrix A consists of all vectors x such that xTA = 0T, where T denotes the transpose of a column vector. The left null space of A is the same as the null space of AT. The left null space of A is the orthogonal complement to the column space of A, and is the cokernel of the associated linear transformation. The null space, the row space, the column space, and the left null space of A are the four fundamental subspaces associated to the matrix A.
Read more about this topic: Kernel (matrix)
Famous quotes containing the words left, null and/or space:
“Thou hast left behind
Powers that will work for thee; air, earth, and skies;
There’s not a breathing of the common wind
That will forget thee; thou hast great allies;
Thy friends are exultations, agonies,
And love, and man’s unconquerable mind.”
—William Wordsworth (1770–1850)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“Not so many years ago there there was no simpler or more intelligible notion than that of going on a journey. Travel—movement through space—provided the universal metaphor for change.... One of the subtle confusions—perhaps one of the secret terrors—of modern life is that we have lost this refuge. No longer do we move through space as we once did.”
—Daniel J. Boorstin (b. 1914)