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Linear Spaces: Row and Column Spaces

In: Matrix Algebra From a Statistician’s Perspective

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  • David A. Harville

    (IBM T.J. Watson Research Center, Mathematical Sciences Department)

Abstract

Associated with any matrix is a very important characteristic called the rank. The rank of a matrix is the subject of Section 4.4. There are several (consistent) ways of defining the rank. The most fundamental of these is in terms of the dimension of a linear space. Linear spaces and their dimensions are discussed in Sections 4.1 through 4.3. Any matrix has two characteristics that are even more basic than its rank; these are two linear spaces that are respectively known as the row and column spaces of the matrix—discussion of row and column spaces is included in the coverage of Sections 4.1 through 4.3. It is shown in Section 4.4 that the column space of a matrix is of the same dimension as its row space; the rank of the matrix equals this dimension. The final section of Chapter 4 (Section 4.5) gives some basic results on the ranks and row and column spaces of partitioned matrices and of sums of matrices.

Suggested Citation

  • David A. Harville, 1997. "Linear Spaces: Row and Column Spaces," Springer Books, in: Matrix Algebra From a Statistician’s Perspective, chapter 4, pages 27-48, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-22677-4_4
    DOI: 10.1007/0-387-22677-X_4
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