Lesson 33 The Kernel of a Linear Transformation.pdf -...

Lesson 33: The Kernel of a Linear Transformation (4.2, 4.3)
Given a linear transformation between two vector spaces, one can ask whether
or not it is one-to-one (defined below). Answering this question can require a
lot of work. In this section, you will learn about the kernel of a linear trans-
formation. It turns out that knowledge about the kernel helps determine if the
transformation is one-to-one.
Definition 1 : A linear transformation T : V → W is one-to-one if for all vectors
v
1
, v
2
∈ V,
T ( v
1
) = T ( v
2
) implies v
1
= v
2
.
Equivalently,
v
1
̸ = v
2
implies T ( v
1
) ̸ = T ( v
2
) .
Thus, T is one-to-one if two distinct vectors in V have two distinct images
in W under T .
Example 1 : Let T : R 2 → R 2 be a linear operator defined by T ( x
1
, x
2
) =
(2 x
1
, x
1
+ x
2
). Show that T is one-to-one.
1

Example 2 : Let T : R 3 → R be a linear transformation defined by
T ( x
1
, x
2
, x
3
) = x
2
.