Orthogonality and Projections
https://www.maa.org/taxonomy/term/42301/all
enA Geometric Intrepretation of the Columns of the (Pseudo) Inverse of A
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-geometric-intrepretation-of-the-columns-of-the-pseudo-inverse-of-a
<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>The pseudo inverse of a matrix \(A\) is used to obtain information about the rows of \(A\).</em></p>
</div></div></div>Obtaining the \(QR\) Decomposition by Pairs of Row and Column Operations
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/obtaining-the-qr-decomposition-by-pairs-of-row-and-column-operations
<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>The author gives an alternative to the usual Gram-Schmidt process and shows how to obtain the “QR Factorization” by using pairs of row column operations.</em></p>
</div></div></div>A Picture is Worth a Thousand Words
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-picture-is-worth-a-thousand-words
<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>The author provides geometric illustrations of four subspaces associated with a matrix. Thinking of a matrix as a map between real vector spaces, the illustrations motivate the decomposition of the domain and range into orthogonal subspaces and the decomposition of vectors into orthogonal components. The author also indicates how the illustrations can be related to the solvability of certain matrix equations.</em></p>
</div></div></div>A Note on the Equality of the Column and Row Rank of a Matrix
https://www.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>Full Rank Factorization of Matrices
https://www.maa.org/programs/faculty-and-departments/course-communities/full-rank-factorization-of-matrices
Online Math Lab at UC Santa Barbara
https://www.maa.org/programs/faculty-and-departments/course-communities/online-math-lab-at-uc-santa-barbara
Projections in real inner product spaces
https://www.maa.org/programs/faculty-and-departments/course-communities/projections-in-real-inner-product-spaces
Gram-Schmidt Calculator
https://www.maa.org/programs/faculty-and-departments/course-communities/gram-schmidt-calculator
The Gram-Schmidt Algorithm
https://www.maa.org/programs/faculty-and-departments/course-communities/the-gram-schmidt-algorithm
Gram-Schmidt Process in Three Dimensions
https://www.maa.org/programs/faculty-and-departments/course-communities/gram-schmidt-process-in-three-dimensions
Gram-Schmidt Process in Two Dimensions
https://www.maa.org/programs/faculty-and-departments/course-communities/gram-schmidt-process-in-two-dimensions
Computing Eigenvalues Using the QR Algorithm
https://www.maa.org/programs/faculty-and-departments/course-communities/computing-eigenvalues-using-the-qr-algorithm
Dot Product Applet
https://www.maa.org/programs/faculty-and-departments/course-communities/dot-product-applet
Terence Tao's Applets
https://www.maa.org/programs/faculty-and-departments/course-communities/terence-taos-applets
Classroom Capsules and Notes for Orthogonality and Projections in Linear Algebra
https://www.maa.org/programs/faculty-and-departments/course-communities/classroom-capsules-and-notes-for-orthogonality-and-projections-in-linear-algebra