From a Question by Dr Purchase
Mr Ottewill
Introduction
In the book we find the conditions stated for a linear transformation, T, as:
(1)
T( k x ) = k T( x )
1 x +
1 x ) + T(
2 x ) = T(
2 x )
(2)
T(
Both of these conditions have a similar feel, namely that if we increase x then T( x ) increases ‘proportionally’. This similar feel leads to the question of whether both conditions are needed.
It is perhaps worth noting that the two condition may be combined into one, namely T( 1 k 1 x + 2 k
1 k T(
1 x ) +
2 x ) =
2 k T( 2 x ) but in a sense there are still two ‘facts’ within this, a multiplicative
element (the k s) and an additive element (the +).
The best way to express the question is perhaps to ask: do transformations exist which satisfy (1) but not (2), or vice versa? If all transformations which satisfy (1) also satisfy (2) then there would presumably be no need to check for (2) as well as (1), and vice versa if all transformations which satisfy (2) also satisfy (1). The answer is not as straightforward as it might seem. As shown below, we can soon show that there are some transformations which satisfy (1) but not (2). It is much harder to find real valued transformations which satisfy (2) but not (1), so hard in fact we cannot explicitly state one, but using the axiom of choice we can show that they exist. The fact that the axiom of choice is a strange thing in itself, as also discussed below, shows how strange the transformation is.
Condition (1) does not imply condition (2)
2
2
y x
x
Consider the transformation T
=
. It can be checked relatively quickly that for
y
2
2
y x
positive k this satisfies (1) but not (2), hence showing that condition (1) by itself is not enough for linearity. (To be more general, you would need to take the positive square root for positive k and the negative root for negative k ).
Condition (2) implies condition (1) for all rational values of k
To show this, we build up the cases of positive integers, then zero and negative integers and finally all rational numbers as follows.
For k Z + :
T(2 x ) = T( x + x ) = T( x ) + T( x ) = 2T( x ) and similarly for T(3 x ) etc.
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