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A Note on the Equality of the Column and Row Rank of a Matrix

by George Mackiw (Loyola College in Maryland)

This article originally appeared in:
Mathematics Magazine
October, 1995

Subject classification(s): Algebra and Number Theory | Linear Algebra
Applicable Course(s): 3.8 Linear/Matrix Algebra

An elementary argument, different from the usual one, is given for the familiar equality of row and column rank. The author takes “full advantage of the following two elementary observations: (1) For any vector \(x\) in \(\mathcal{R}^n\) and matrix \(A\), \(Ax\) is a linear combination of the columns of \(A\), and (2) vectors in the null space of \(A\) are orthogonal to vectors in the row space of \(A\), relative to the usual Euclidean product.”

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Capsule Course Topic(s):
Linear Algebra | Orthogonality and Projections
Linear Algebra | Rank of Matrices
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