Rank of Matrices
https://maa.org/taxonomy/term/42329/all
enWhen is Rank Additive?
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/when-is-rank-additive
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This capsule presents necessary and sufficient conditions for the matrix rank of a sum to be the sum of the ranks. The crux of the argument uses the fact that the rank of a matrix is the size of its largest invertible submatrix.</em></p>
</div></div></div>A Note on the Equality of the Column and Row Rank of a Matrix
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-note-on-the-equality-of-the-column-and-row-rank-of-a-matrix
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>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.”</em></p>
</div></div></div>Row Rank Equals Column Rank
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/row-rank-equals-column-rank
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A short elementary proof of the equality of row rank and column rank is given. The proof requires only the definition of matrix multiplication and the fact that a minimal spanning set is a basis.</em></p>
</div></div></div>A Direct Proof That Row Rank Equals Column Rank
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-direct-proof-that-row-rank-equals-column-rank
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A row (column) of a matrix is called “extraneous” if it is a linear combination of the other rows (columns). The author shows that deleting an extraneous row or column of a matrix does not affect the row rank or column rank of a matrix. This fact establishes the theorem in the title.</em></p>
</div></div></div>Classroom Capsules and Notes for under Rank of Matrices in Linear Algebra
https://maa.org/programs/faculty-and-departments/course-communities/classroom-capsules-and-notes-for-under-rank-of-matrices-in-linear-algebra