Relation To The Row Space
Let A be an m by n matrix (i.e., A has m rows and n columns). The product of A and the n-dimensional vector x can be written in terms of the dot product of vectors as follows:
Here a1, ..., am denote the rows of the matrix A. It follows that x is in the null space of A if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (because if the dot product of two vectors is equal to zero they are by definition orthogonal).
The row space of a matrix A is the span of the row vectors of A. By the above reasoning, the null space of A is the orthogonal complement to the row space. That is, a vector x lies in the null space of A if and only if it is perpendicular to every vector in the row space of A.
The dimension of the row space of A is called the rank of A, and the dimension of the null space of A is called the nullity of A. These quantities are related by the equation
The equation above is known as the rank–nullity theorem.
Read more about this topic: Kernel (matrix)
Famous quotes containing the words relation to the, relation to, relation, row and/or space:
“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“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)
“The church is a sort of hospital for men’s souls, and as full of quackery as the hospital for their bodies. Those who are taken into it live like pensioners in their Retreat or Sailor’s Snug Harbor, where you may see a row of religious cripples sitting outside in sunny weather.”
—Henry David Thoreau (1817–1862)
“And Space with gaunt grey eyes and her brother Time
Wheeling and whispering come,”
—James Elroy Flecker (1884–1919)