MG Week 12 - 13

Linear Transformations

Linear Transformations from Rⁿ to R^m

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

Let T : R² → R³ be defined by

∴ T ° S : R³ → R³ is

Geometric Linear Transformations

Scalings in R²

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

Reflection about the y-axis

Reflection about the line y = x

Reflection about the line y = x tan θ

Rotations in R²

Anti-clockwise rotation through an angle θ

Translations

Not a linear transformation

Shears

Shear in the x-direction by a factor of k

Shear in the y-direction by a factor of k

fin