MATH136 Lecture Notes - Lecture 14: Transformation Matrix, Linear Map, Linear Combination
Wednesday, May 31
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Lecture 14 : Linear mappings (transformations) I (Refers to
3.2)
Concepts:
1. linear mapping, transformation, operator
2. matrix viewed as a linear mapping
14.1. Definition
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A function T : ℝn → ℝm is said to be a linear mapping (also called
“linear transformation”) if, for any pair of vectors u and v in ℝn and scalars α and β,
T(αu + βv) = αT(u) + βT(v)
We sometimes say that T is linear if it “respects linear combinations”. If T : ℝn → ℝn is
a linear mapping the we day T is a linear operator on ℝn.
Note that, when viewed together, the following two conditions are equivalent to the one
condition given above:
1) T(αx) = αT(x) (We say that T “preserves” scalar multiples.)
2) T(x + y) = T(x) + T(y) (We say that T “preserves” addition.)
Sometimes it is simpler to show that the function T preserves scalar multiples and
preserves addition separately.
14.1.1 Example – We define the function T : ℝ3 → ℝ2, as
T [ (x1, x2, x3) ] = (x1, x2)
For example, T [ (1, 5, 2) ] = (1, 5). This is a linear mapping since
T [α(x1, x2, x3) + β(y1, y2, y3) ] = T [(αx1, αx2, αx3) + (βy1, βy2, βy3) ]
= T [(αx1 + βy1, αx2 + βy2, αx3+ βy3) ]
= (αx1 + βy1, αx2 + βy2)
= (αx1 , αx2 ) + (βy1, βy2)
= α(x1 , x2 ) + β(y1, y2)
= αT [(x1 , x2, x3 ) ] + βT [(y1, y2, y3) ]
14.2 The matrix Am × n viewed as a linear mapping from ℝn into ℝm.
Given a matrix Am × n = [aij] m × n we define a function T : ℝn → ℝm as T(x) = Ax.
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Document Summary
Wednesday, may 31 lecture 14 : linear mappings (transformations) i (refers to. Concepts: linear mapping, transformation, operator, matrix viewed as a linear mapping is said to be a linear mapping (also called. Definition a function t : n m. Linear transformation ) if, for any pair of vectors u and v in n and scalars and , We sometimes say that t is linear if it respects linear combinations . If t : n n a linear mapping the we day t is a linear operator on n. Note that, when viewed together, the following two conditions are equivalent to the one condition given above: T( u + v) = t(u) + t(v) is: t( x) = t(x) (we say that t preserves scalar multiples. , t(x + y) = t(x) + t(y) (we say that t preserves addition. ) Sometimes it is simpler to show that the function t preserves scalar multiples and preserves addition separately.