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Lecture 18

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University of Waterloo
MATH 136
Robert Sproule

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 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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