Linear Algebra

Topic: Geometry of Linear Maps 277
To finish, we will consider a familiar application, in calculus. On the left
below is a picture, like the ones that started this Topic, of the action of the nonlinear
function y(x) =x 2 + x. As described at that start, overall the geometric
effect of this map is irregular in that at different domain points it has different
effects (e.g., as the domain point x goes from 2 to −2, the associated range point
f(x) at first decreases, then pauses instantaneously, and then increases).
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But in calculus we don’t focus on the map overall, we focus on the local effect
of the map. The picture on the right looks more closely at what this map does
near x =1. Atx = 1 the derivative is y ′ (1) = 3, so that near x =1wehave
that ∆y ≈ 3 · ∆x; in other words, (1.001 2 +1.001) − (1 2 +1)≈ 3 · (0.001). That
is, in a neighborhood of x = 1, this map carries the domain to the codomain by
stretching by a factor of 3 — it is, locally, approximately, a dilation. This shows
a small interval in the domain (x − ∆x..x+∆x) carried over to an interval in
the codomain (y − ∆y..y+∆y) that is three times as wide: ∆y ≈ 3 · ∆x.
x =1
(When the above picture is drawn in the traditional cartesian way then the prior
sentence is usually rephrased as: the derivative y ′ (1) = 3 gives the slope of the
line tangent to the graph at the point (1, 2).)
Calculus considers the map that locally approximates the change ∆x ↦→
3 · ∆x, instead of the actual change map ∆x ↦→ y(1 + ∆x) − y(1), because the
local map is easy to work with. Specifically, if the input change is doubled, or
tripled, etc., then the resulting output change will double, or triple, etc.
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y =2
3(r ∆x) =r (3∆x)
(for r ∈ R) and adding changes in input adds the resulting output changes.
3(∆x1 +∆x2) =3∆x1 +3∆x2
In short, what’s easy to work with about ∆x ↦→ 3 · ∆x is that it is linear.
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