# Mmt - Linear Algebra Notes Exercises and Lecture Mondays...

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Linear Algebra: Notes, Exercises, and Lecture Monday ʼ s start on problem 15.5 (from Applications of Linear Algebra by Anton and Rorres) was atrocious. The whole point of abstraction is to avoid tedious repetition. What I was about to do (that is, what you were about to let me do) amounted to a reproof for the associativity of matrix multiplication by reversing the order of summation. A coordinateless proof that illustrates the power of the notation of linear algebra follows a few observations: 1. The product of an pxq matrix A with a qx1 column vector is a linear combination of the columns of A. (This means that the range of a linear transformation with matrix A is the span of A ʼ s columns.) Be convinced: a 11 a 12 a 1 q a 21 a 22 a 2 q a p 1 a p 2 a pq b 1 b 2 b q = a 11 b 1 + a 12 b 2 + + a 1 q b q a 21 b 1 + a 22 b 2 + + a 2 q b q a p 1 b 1 + a p 2 b 2 + + a pq b q = b 1 a 11 a 21 a p 1 + b 2 a 12 a 22 a p 2 + + b q a 1 q a 2 q a pq

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