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 14, the following proof uses the associativity of matrix multiplication (often and
without mention) and the fact that a set of nonzero 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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 Spring '10
 N.Darnton
 Physics, Linear Algebra, Vector Space, Linear combination, linear transformation

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