普通地质学3.7 Row space and column space of a matrix普通地质学.pdf

Row space and column space of a matrixof a matrix Introduction If A is an m ×n matrix, each row of A is an n-tuple of real numbers and hence can be considered as a vector in n R . The m vectors corresponding to the rows of A will be referred to as the row vectors of A. Similarly, each column of ASimilarly, each column of A can be considered ascan be considered as a vector in Rm, and we can associate n column vectors with the matrix A. Introduction In this section we define row space and column space of a matrix and show one example. AAnd we discuss the nd we discuss the properties of row/column space.properties of row/column space. Outline 1. Definitions of row space and column space of a matrix 2. Example 3. Properties of row/column space Definitions of row and column space of a matrix If A is an m ×n matrix, the subspace of Rm spanned by the column vectors of A is called the column space of A, denoted by R(A). The subspace of The subspace of RRnn spanned by the row vectors of A is spanned by the row vectors of A is T called the row space of A, denoted by R(A ). The rank of a matrix A, denoted rank(A), is the dimension of the row space of A, namely T rank(A) = dimR(A ) . Example Ex. Given a 2 ×3 matrix 1 0 0 A  . 0 1 0 Find the row space and the column space of A. Solution. Solution. The row space of A isThe row space of A is  1 0        T        R(A )    0  1