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MG Week 12 - 13
Linear Transformations
Linear Transformations from Rn to Rm
Conditions for Linear Transformation
- T(0) = 0
- T(cv) = c T(v)
- T(u + v) = T(u) + T(v)
If any of the conditions are not met, T is not a linear transformation.
Composite Functions
S ° T is the composition of S with T
(S ° T)(u) = S(T(u))
T ° S is the composition of T with S
(T ° S)(u) = T(S(u))
Note: S ° T ≠ T ° S
Example
Let S : R3 → R2 be defined by
S
=
Let T : R2 → R3 be defined by
T
=
∴ T ° S : R3 → R3 is
(T ° S)
= T
S
= T
=
=
0x + 0y + z
0x + 0y + z
x + y + 0z
=
∴ Standard Matrix for T ° S is
Geometric Linear Transformations
Scalings in R2
Standard matrix :
Effect: To scale by a factor of λ1 along the x-axis, and by a factor of λ2 along the y-axis
Special Case: λ1 = λ2 = λ
λ > 1 → Effect: A dilation by a factor of λ
0 < λ < 1 → Effect: A contraction by a factor of λ
Reflections in R2
Reflection about the x-axis
Standard matrix :
Reflection about the y-axis
Standard matrix :
Reflection about the line y = x
Standard matrix :
Reflection about the line y = x tan θ
Standard matrix :
cos 2θ sin 2θ
sin 2θ -cos 2θ
Rotations in R2
Anti-clockwise rotation through an angle θ
Standard matrix :
Translations
Not a linear transformation
Shears
Shear in the x-direction by a factor of k
Standard matrix :
Shear in the y-direction by a factor of k
Standard matrix :
fin
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