Dimension
The dimension of the column space is called the rank of the matrix. The rank is equal to the number of pivots in the reduced row echelon form, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4 × 4 matrix in the example above has rank three.
Because the column space is the image of the corresponding matrix transformation, the rank of a matrix is the same as the dimension of the image. For example, the transformation R4 → R4 described by the matrix above maps all of R4 to some three-dimensional subspace.
The nullity of a matrix is the dimension of the null space, and is equal to the number of columns in the reduced row echelon form that do not have pivots. The rank and nullity of a matrix A with n columns are related by the equation:
This is known as the rank-nullity theorem.
Read more about this topic: Column Space
Famous quotes containing the word dimension:
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)
“Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.”
—Shoshana Zuboff (b. 1951)
“By intervening in the Vietnamese struggle the United States was attempting to fit its global strategies into a world of hillocks and hamlets, to reduce its majestic concerns for the containment of communism and the security of the Free World to a dimension where governments rose and fell as a result of arguments between two colonels’ wives.”
—Frances Fitzgerald (b. 1940)