Chapter 18

Linear FunctionsCalculus of many variables involves the consideration of functions of many variables, justas calculus of one variable, considered earlier is about functions of one variable. Recallthis could involve a function which has vector values, but the function depended on onlyone variable. When you consider functions of many variables, the easiest are those whichare linear.

18.1 The Matrix of a Linear TransformationThe next definition is on what it means for a function to be linear.

Definition 18.1.1 Let T : Rn → Rm. This function is linear if whenever α,β arenumbers and x,y are vectors in Rn, it follows that

T (αx+βy) = αT (x)+βT (y)

Such linear functions are also called linear transformations or linear maps. Also, forT : Rn → Rm linear, it is standard to write T ∈ L (Rn,Rm).

The first thing we need to do is to give an easily usable description of such a linearfunction. This will involve special vectors called ek. Also, from now on bold face x willrefer to a vector as earlier but now the vector will be written as a column vector. Thus

x=

 x1...

xn

would denote a vector in Rn. Actually, to save space, this vector will often be written as(

x1 · · · xn)T the exponent T indicating that one is to make this row of numbers into a

column of numbers as above. All other conventions about addding and and multiplying bynumbers (scalars) are the same as discussed earlier. Now for the definition of the specialvectors ek,

Definition 18.1.2 ek is the vector(

0 · · · 1 · · · 0)T where there is a 1 in

the kth position and a 0 in every other position. Thus if x=(

x1 · · · xn)T is a vector

in Rn,x= x1e1 + x2e2 + · · ·+ xnen

Written out, this is of the form x1...

xn

= x1

 1...0

+ · · ·+ xn

 0...1

As an example,  1

23

= 1

 100

+2

 010

+3

 001

Be sure you understand this before reading further. You need to use the rules of vector ad-ddition and scalar multiplication discussed earlier but this time applied to column vectors.

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