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Let r dim U, and let S represent any set of r linearly independent matrices that is a basis for U. Suppose that U ⊂ V, in which case all r of the matrices in S are in V. 9) S is a basis for V, implying that V sp(S) U. D. It follows from the discussion of Subsection b that dim(Rm×n ) mn . 3). f. Extracting a basis from a spanning set A basis for a linear space can be extracted from any spanning set, as indicated by the following theorem. 11. Any set S that spans a linear space V (of m × n matrices) contains a subset that is a basis for V.

4. There are several (consistent) ways of defining the rank. The most fundamental of these is in terms of the dimension of a linear space. 3. 3. 4 that the column space of a matrix is of the same dimension as its row space; the rank of the matrix equals this dimension. 5) gives some basic results on the ranks and row and column spaces of partitioned matrices and of sums of matrices. 1 Some Definitions, Notation, and Basic Relationships and Properties a. Column spaces The column space of an m × n matrix A is the set whose elements consist of all m-dimensional column vectors that are expressible as linear combinations of the n columns of A.

A submatrix of an n × n matrix is called a principal submatrix if it can be obtained by striking out the same rows as columns (so that the ith row is struck out whenever the ith column is struck out, and vice versa). The r × r (principal) submatrix of an n × n matrix obtained by striking out its last n − r rows and columns is referred to as a leading principal submatrix (r 1, . . , n). A principal submatrix of a symmetric matrix is symmetric, a principal submatrix of a diagonal matrix is diagonal, and a principal submatrix of an upper or lower triangular matrix is respectively upper or lower triangular, as is easily verified.