In this lecture on matrix spaces, rank 1 matrices, and small world graphs, the concept of vector spaces is extended to include matrices and differential equations. The rank of matrices, particularly rank 1 matrices, is discussed in detail, and it is shown that every rank 1 matrix can be written as a row times a column. The lecture also explores examples of subspaces, their dimensions, and bases, including those of symmetric and upper triangular matrices. Finally, the idea that rank 1 matrices are the building blocks for all matrices is introduced.
Matrix Spaces, Rank 1 matrices, and Small World Graphs -- Lecture 11. Some more discussion of matrices, bases, dimensions, and vector spaces. Rank of matrices (and rank 1 matrices) are also discussed.
Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 22, 2008). License: Creative Commons BY-NC-SA.
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