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

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Linear Algebra: Notes, Exercises, and Lecture Monday’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’s columns.) Be convinced: 2. The dot product of n x 1 column vectors v and w is the matrix product v t w. 3. If A and B can be multiplied, then (AB) t = B t A t . (Check that the shapes are right.) 4. vv 0 and . vv = 0 iff v = 0. In addition to using 1-4, the following proof uses the associativity of matrix multiplication (often and without mention) and the fact that a set of non-zero vectors is linearly independent iff the only linear combination of the vectors equal to 0 is the trivial one where all coefficients are 0. Here we go:

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

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