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Lecture 18

# Lecture 18.pdf

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University of Waterloo

Mathematics

MATH 136

Robert Sproule

Winter

Description

Friday, February 14 − Lecture 18 : Operations on mappings
Concepts:
1. Algebraic operations onlinear transformation
n m n m
18.1 Definition − Let T: ℝ → ℝ and L : ℝ → ℝ be two linear mapping. The
mappings T and L are said to be equal mappings if and only if T(x) = L(x) for all x in
their domain. Addition of mappings T + L is defined as
(T + L)(x) = T(x) + L(x)
Scalar multiplication of mapping αT, for any scalar α, is defined as
(αT)(x) = αT(x)
n m n m
18.2 Theorem − Let T : ℝ → ℝ and L : ℝ → ℝ be two linear mapping. Let A and B
be the matrices induced byT and L respectively. Then the matrix C = A + B is the unique
matrix induced by the linear mapping T + L and the matrix αA is the unique matrix
induced by the linear mapping a αT.
Properties linear mappings algebra.
n m
18.3 Theorem − Let L denote the set of all linear mappings from ℝ to ℝ . Then
1) Closure properties:L is closed under addition and scalar multiplication.
2) Addition properties:
i. Addition in L is associative and commutative.
ii. There exists inL a linear mapping 0 such that L + 0 = L for allLL.in
iii. For every L inL there exist a mapping –L in L such that L + −L = 0.
3) Scalar multiplication properties:
i. Linear mappings in L distri

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