Page 10 - 35Linear Algebra
P. 10
10 What is Linear Algebra?
But lets think carefully; what is the left hand side of this equation doing?
Functions and equations are different mathematical objects so why is the
equal sign necessary?
A Sophisticated Review of Functions
If someone says
“Consider the function of two variables 7β − 13b.”
we do not quite have all the information we need to determine the relationship
between inputs and outputs.
Example 1 (Of organizing and reorganizing information)
You own stock in 3 companies: Google, Netflix, and Apple. The value V of your
stock portfolio as a function of the number of shares you own s N , s G , s A of these
companies is
24s G + 80s A + 35s N .
1
2 ?
Here is an ill posed question: what is V
3
The column of three numbers is ambiguous! Is it is meant to denote
• 1 share of G, 2 shares of N and 3 shares of A?
• 1 share of N, 2 shares of G and 3 shares of A?
Do we multiply the first number of the input by 24 or by 35? No one has specified an
order for the variables, so we do not know how to calculate an output associated with
a particular input. 1
A different notation for V can clear this up; we can denote V itself as an ordered
triple of numbers that reminds us what to do to each number from the input.
1
Of course we would know how to calculate an output if the input is described in
the tedious form such as “1 share of G, 2 shares of N and 3 shares of A”, but that is
unacceptably tedious! We want to use ordered triples of numbers to concisely describe
inputs.
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