100 010 The row space of A is the set of all 3-tuples of the form a(100)+B(010)=(aB0) The column space of A is the set of all vectors of the form 0 0 1+0 Thus the row space of A is a two-dimensional subspace of Rx3 and the column space of A is R Theorem The column space and the row space of a matrix have the same dimension Definition The column(row) rank of a matrix is the dimension of the vector space that is spanned by its columns(rows) In short from this definition we know that the column rank is the number of linearly independent column of a matrix Theorem The column rank and row rank of a matrix are equal, that is rank(A)=rank(A')< min(number of rows, numbers of columns) Definition A full(short)rank matrix is a matrix whose rank is equal(fewer)to the number of columns it contains ome useful result rank(AB)< min(rank(A), rank(B))Let A =  1 0 0 0 1 0  . The row space of A is the set of all 3-tuples of the form α(1 0 0) + β(0 1 0) = (α β 0). The column space of A is the set of all vectors of the form α  1 0  + β  0 1  + γ  0 0  . Thus the row space of A is a two-dimensional subspace of R 1×3 and the column space of A is R 2 . Theorem: The column space and the row space of a matrix have the same dimension. Definition: The column(row) rank of a matrix is the dimension of the vector space that is spanned by its columns (rows). In short from this definition we know that the column rank is the number of linearly independent column of a matrix. Theorem: The column rank and row rank of a matrix are equal, that is rank(A)=rank(A’)≤ min(number of rows, numbers of columns). Definition: A full (short) rank matrix is a matrix whose rank is equal (fewer) to the number of columns it contains. Some useful results: 1. rank(AB)≤ min(rank(A),rank(B)). 12