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:
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)
“The base emotions Plato banned
have left a radio-active and not radiant land.”
—Libby Houston (b. 1941)
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“If we remembered everything, we should on most occasions be as ill off as if we remembered nothing. It would take us as long to recall a space of time as it took the original time to elapse, and we should never get ahead with our thinking. All recollected times undergo, accordingly, what M. Ribot calls foreshortening; and this foreshortening is due to the omission of an enormous number of facts which filled them.”
—William James (1842–1910)