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Lecture 13

# MATA23H3 Lecture Notes - Lecture 13: Linear Map, Row And Column Vectors, Transformation MatrixPremium

Department
Mathematics
Course Code
MATA23H3
Professor
Chrysostomou( G)
Lecture
13

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MATA23 - Lecture 13 - Linear Transformations
Linear Systems
A function is a rule that transforms one real number into another
Example: Let f(x)=2x, and g(x) = x2, x R
Consider a linear system: A~x =~y, where ~x Rnand ~y Rm
Let A=
2 3
1 0
12
2 3
1 0
12
x1
x2=
y1
y2
y3
A1
0=
2 3
1 0
12
1
0=
2
1
4
A1
1=
2 3
1 0
12
1
1=
5
1
2
A0
0=
0
0
0
1
0+1
1=2
1
A2
1=
2 3
1 0
12
2
1=
7
2
6
TA: the matrix transformation that maps a 2×1column vector ~x in R2into the 3×1column
vector TA(~x) = A~x in R3(see diagram 1)
In component form: TA(~x) = TA(x1
x2) =
2 3
1 0
12
x1
x2=
2x1+ 3x2
x1
4x12x2
Transformations
A transformation (or function) Tfrom Rnto Rmis a rule that assigns to each vector ~x in Rn
into a vector T(~x)in Rm, where:
Rnis the domain of T,
Rmis the codomain of T,
T(~x)is the image of ~x under T,
1