Transcription of Section 3.3. Matrix Rank and the Inverse of a Full Rank Matrix
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Section 3.3. Matrix Rank and the Inverse of a Full Rank Matrix
Matrix rank and the Inverse of a Full rank Matrix1 Section rank and the Inverseof a Full rank lengthy Section (21 pages in the text) gives a thorough study of therank of a Matrix (and Matrix products) and considers inverses of matrices brieflyat the that the row space of a matrixAis the span of the row vectors ofAand the row rank ofAis the dimension of this row space. Similarly, the columnspace ofAis the span of the column vectors ofAand the column rank is thedimension of this column space. You will recall that the dimension of the columnspace and the dimension of the row space of a given Matrix are the same (seeTheorem of Fraleigh and Beauregard sLinear Algebra, 3rd Edition, Addison-Wesley Publishing Company, 1995, The rank of a Matrix ). We now give aproof of this based in part on Gentle s argument and on Peter Lancaster sTheoryof Matrices, Academic Press (1969), page 42. First, we need a {ai}ki=1={[ai1, ai2, .. , ain]}ki=1be a set of vectors inRnand let Sn.
Section 3.3. Matrix Rank and the Inverse of a Full Rank Matrix Note. The lengthy section (21 pages in the text) gives a thorough study of the rank of a matrix (and matrix products) and considers inverses of matrices briefly at the end. Note. Recall that the row space of a matrix A is the span of the row vectors of
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