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Chapter4 - Chapter 4 Vector Spaces Associated with Matrices...

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Chapter 4 Vector Spaces Associated with Matrices Section 4.1 Row Spaces and Column Spaces Discussion 4.1.1 Each m × n matrix is naturally associated with three vector spaces, namely, the row space, the column space and the nullspace. These three vector spaces provide us with insights into the relationships between solutions of linear systems and the coefficient matrix. In the following, we shall first discuss the row and column spaces of a matrix. Definition 4.1.2 Let A = ( a ij ) be an m × n matrix, i.e. A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . a m 1 a m 2 · · · a mn . The row space of A is the subspace of R n spanned by the rows of A . The column space of A is the subspace of R m spanned by the columns of A . Let r 1 , r 2 , · · · , r m be the m rows of A , i.e. r 1 = ( a 11 , a 12 , . . . , a 1 n ) , r 2 = ( a 21 , a 22 , . . . , a 2 n ) , . . . r m = ( a m 1 , a m 2 , . . . , a mn ) , and let c 1 , c 2 , · · · , c n be the n columns, i.e.
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2 Chapter 4. Vector Spaces Associated with Matrices c 1 = a 11 a 21 . . . a m 1 , c 2 = a 12 a 22 . . . a m 2 , · · · , c n = a 1 n a 2 n . . . a mn . Then the row space of A = span { r 1 , r 2 , · · · , r m } and the column space of A = span { c 1 , c 2 , · · · , c n } . Remark 4.1.3 The row space of A is the same as the column space of A T while the column space of A is the same as the row space of A T . Example 4.1.4 1. Consider the matrix A = 2 - 1 0 1 - 1 3 - 5 1 0 1 0 1 . The rows of A are r 1 = (2 , - 1 , 0) r 2 = (1 , - 1 , 3) r 3 = ( - 5 , 1 , 0) r 4 = (1 , 0 , 1) and its columns are c 1 = 2 1 - 5 1 , c 2 = - 1 - 1 1 0 , c 3 = 0 3 0 1 . Note that the row space of A is a subspace of R 3 while the column space of A is a subspace of R 4 . 2. Find a basis for the row space and a basis for the column space of A given in Part 1. Hence state the dimension of the row space and the dimension of the column space of A . Solution (In Remark 4.1.9 and Remark 4.1.13, we have methods for finding bases for row and column spaces of matrices. For this example, we just do it by brute
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Section 4.1. Row Spaces and Column Spaces 3 force.) We first consider the row space of A . The row space of A is a subspace of R 3 and hence any basis of the row space of A contains at most three vectors. Since the rows r 1 , r 2 and r 3 of A are linearly independent (check it), we see that they form a basis for the row space of A . All three columns c 1 , c 2 and c 3 of A are linearly independent (check it) and so form a basis for the column space of A . In this example, both the row space and the column space of A have the same dimension, which is 3. (In general, both the row space and the column space of a matrix always have the same dimension.) Discussion 4.1.5 We say that two matrices A and B are row equivalent if one can be obtained from the other by a series of elementary row operations. Using the concept of row equivalent matrices, we shall develop methods to find bases for row spaces and column spaces. First, let us note some observations:
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