Applied Mathematics 1411A/B Lecture Notes - Lecture 3: Augmented Matrix, Lincoln Near-Earth Asteroid Research, Elementary Matrix
12 Mar 2015
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More on linear systems and invertible matrices (1. 6; pg. Recall: last day, we learned about the concept of inverting a matrix. Theorem: every system of linear equations has either no solution, exactly one solution, or infinitely many solutions. Proof (that if more than one solution, then infinitely many): Theorem: if a is an invertible of equations nn matrix, then for each column vector b, the system bx a has exactly one solution, namely x b. Question: say you wanted to solve all the systems where a is the same for all systems. Idea: if a is invertible, then augmented matrix system . Theorem: let a be a square matrix: if b is a square matrix satisfying, if b is a square matrix satisfying. 0x a has only the trivial solution nn matrix, then the following statements are: a is invertible, the homogeneous system, the reduced row-echelon from of a is, a is expressible as a product of elementary matrices.
