Applied Mathematics 1411A/B Chapter Notes - Chapter 4.9: Euclidean Vector, Orthogonal Complement, Transformation Matrix
30 Dec 2018
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If a is an m x n matrix, then: a) b) The null space of a and the row space of a are orthogonal complements in rn. The null space of at and the column space of a are orthogonal complements in rn. Orthogonal complements being when all the physical vectors in a set are directly perpendicular to all the physical vectors in another set. This is the link between physical vectors in a cartesian plane and a matrix. These spaces in rn which are completely dependant on the matrix values, those have these certain relationship which allows us to predict geometric appearance. In the last section we established a fundamental link between a matrix and sets of geometric vectors in rn. (recall the orthogonality of vectors between certain vector sets defined by a matrix?). In this section, we will be studying the effects of transforming that matrix and how it affects the geometric representation of the set in rn.
