So far in your calculus career you’ve seen derivatives of real- valued
functions of a single variable
(y = f(x)) and also vector-valued functions of a single variable (r(t) = hx(t),
y(t), z(t)i). You’ve
also seen “partial derivatives” of real-valued functions of several variables (w
= f(x, y, z)). It’s now
time to deal with the general case of functions from Rm to Rn, i.e., functions
which take points in
Rm as input and and have points in Rn as output. Before we can do that, however,
it’s best to
discuss matrices and linear function s .
An m× n matrix is a “rectangular array” of m·n things: numbers, variables,
what have you.
For example,
is a 2 × 3 matrix of numbers. It has 2 rows: (1 2 3) and (2 3 4), and 3
columns:
and
Note that we typically write vectors in R2 or R3 as 1 × 2 or 1 × 3 matrices which
we call row
vectors: (a b) or (a b c), though we could just as well write them as column
vectors:
You multiply two matrices in the fol lowing manner . If A is an m×n matrix and
B is an n×p
matrix (it’s crucial that the number of columns of A match the number of rows of
B), then AB
is the m × p matrix you get by taking the dot products of the rows of A with the
columns of B.
There are m such rows of A and p such columns of B, yielding the m · p entries
of AB.
Example 1:
Then AB is just (−6), the dot product of the one row of A (in R3)
with the one column of B (again
in R3).
Example 2:
Adding matrices is simpler . We add two matrices of the same dimensions (they
must have the
same number of rows and columns) component-wise.
Example 3:
Exercises: Let
1. Compute, if possible, AB, BA, AC, CA, BC, and CB.
2. Compute, if possible, A + B, A + C, B + C, AB + C, and CA − A.
A function from Rm to Rn takes vectors in Rm to
vectors in Rn. We’ll think of both the input
and output vectors here as column vectors. This may seem strange at first. Why
not row vectors?
After all, that’s the way we’ve been writing them. The reason is that we’ve also
been writing our
function notation with the “function” on the left (we write f(x), not (x)f).
We’ll see later how
that plays out with matrices.
A linear function L from Rm to Rn is a function of the
form
L(x) = Ax + b,
where b = L(0) is a (column) vector in Rn and A is an n×m matrix,
called the coefficient matrix
of L. Here you view x as a column vector (i.e., an m×1 matrix) of variables and
Ax is the product
of an n×m and an m×1 matrix. I’m using a bold face L notation here since the
function’s output
values are vectors.
Example 4:
So, for example
Note that the linear function L in example 4 is made up of 3 real-valued
linear functions of
two variables :
This is what happens in general: a function f from Rm to Rn
is a (column) vector of n
real-valued functions of m variables. (Again I’ve used a bold face f since the
output is a vector.)
Example 5:
is a function from R2 or R3 made up from the 3
real-valued functions f1(x, y) = 3xey−7, f2(x, y) =
x+ln y, fw(x, y) = 2y. (Sorry, I reverted to our old notation and wrote the
input as a row vector.)
Why are linear functions so special? Well, they’re
obviously rather simple; no high powers ,
sine functions, etc... But what really sets them apart is their predictability.
Specifically, how the
output changes is easily computed from the change in the input.
Let’s use the typical notation
for the change in a function’s output and
for the change in the input. Then for a
linear function L with a corresponding
coefficient matrix A we have
In fact, suppose you have a function f from Rm to Rn
with the following properties : 1)
f (0) = 0 (here the first 0 is the vector of all zeros in R m and the
second 0 is the vector of all
zeros in Rn); 2) for any two vectors x and y in Rm and any
two real numbers (scalars) a and b,
f (ax + by) = af(x) + bf(y). Then it turns out that f must be a linear function.
Let’s look at a special case. Suppose we have a function f
from R3 or R2 that has the properties
above. Suppose
Then you should convince yourself that f (x) = Ax where A
is the 2 × 3 matrix from the exercises
above.
Exercises: (Use the same matrices A, B and C as in
the previous exercises .)
3. Let g be the linear function from R2 or R3
defined by g(x) = Bx. Find g(i) and g(j). Do
the same for the linear function h from R2 or R2 defined
by h(x) = Cx.
4. Let f be the linear function above. Which of the
following compositions make sense:
5. For each of the compositions in exercise 4 that make
sense, find the composite function.
6. Which of the compositions in exercise 5 are linear functions? What are the
associated
coefficient matrices?
