Wednesday, February 5 − Lecture 14 : Linear mappings (transformations) I
Concepts:
1. linear mapping (transformation)
2. matrix viewed as a linear mapping
14.1. Definition − A linear mapping (also called “linear transformation”) is a function
T : ℝ → ℝ where, for any pair of vectors uand v in ℝ and scalars α and β,
T(αu + βv) = αT(u) + βT(v)
We sometimes say that T is linear if it “respects linear combinations”.
Note that some authors separate the definition of a linear transformation in two parts:
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.)
These definitions are equivalent.
14.1.1 Example – We define the function T: ℝ → ℝ , as 2
T [ (x , x , x )] = (x , x )
1 2 3 1 2
For example, T [ (1, 5, 2) ]= (1, 5). This is a linear mapping since
T [α(x1, x2, x3) + β(y 1 y2, y3)] = T [(αx 1 αx 2 αx 3 + (βy ,1βy ,2βy )3 ]
= T [(αx 1 βy ,1αx + 2y , αx2+ βy 3 3 ]
= (αx +1βy , 1x + 2y ) 2
= (αx 1, αx2) + (βy , 1y ) 2
= α(x ,1x )2+ β(y , y 1 2
= αT [(x1, x2, 3 ) ] + βT [(y 1 y2, y3)]
14.2 The matrix A m × niewed as a linear mappingfrom ℝ into ℝ . m
Given a matrix A m × n= [aij m × n with rows r,iwe define a function T : ℝ → ℝ as m
T(x) = Ax.
In fact T : ℝ → ℝ is a well-defined function. We claim it is a linear mapping. n
For two vectors x and y in ℝ and any scalars α and β we have:
- Hence the matrix A m × ncan be viewed as a functionmapping a vector x in ℝ to a n
vector Ax in ℝ .m
- We have also shown that matrices distributes over a linear combination. That is

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