Example 2. Consider the group R² of vectors for r, y real. This is a group under com- () - () -G) b' we have (x+u` ponentwise addition: u + a Given a 2 x 2 matrix M= \y+ v/ a group homomorphism Lm:R² →R² defined by ах + by' cк+ dy a LM = The kernel of Lm is called the nullspace of the matrix M (or the linear function LM) in linear algebra. See Definition 7.2.3. Exercise 3.5.3 Show that Lm in example 2 is indeed a group homomorphism. What is its kernel?
Example 2. Consider the group R² of vectors for r, y real. This is a group under com- () - () -G) b' we have (x+u` ponentwise addition: u + a Given a 2 x 2 matrix M= \y+ v/ a group homomorphism Lm:R² →R² defined by ах + by' cк+ dy a LM = The kernel of Lm is called the nullspace of the matrix M (or the linear function LM) in linear algebra. See Definition 7.2.3. Exercise 3.5.3 Show that Lm in example 2 is indeed a group homomorphism. What is its kernel?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Example 2. Consider the group R² of vectors
for r, y real. This is a group under com-
u
+
(x+u`
a
Given a 2 x 2 matrix M=
b'
we have
ponentwise addition:
\y+ v/
a group homomorphism Lm:R² →R² defined by
ах + by'
cк+ dy
a
LM
=
The kernel of Lm is called the nullspace of the matrix M (or the linear function LM) in linear
algebra. See Definition 7.2.3.
Exercise 3.5.3 Show that Lm in example 2 is indeed a group homomorphism. What is its
kernel?
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