Ch 1.8 - Introduction to Linear Transformations

Flashcards covering key vocabulary and concepts from the 'Introduction to Linear Transformations' lecture.

Transformation (or Function or Mapping)

A rule that assigns to each vector in Rⁿ, called the domain of T, a vector in Rᵐ, called the codomain of T.

Domain of T

The set Rⁿ from which a transformation T maps vectors.

Codomain of T

The set Rᵐ into which a transformation T maps vectors.

Image of x

For a vector x in Rⁿ, the vector T(x) in Rᵐ is called the image of x under the action of T.

Range of T

The set of all images T(x) from vectors in the domain of T.

Matrix Transformation

A mapping where T(x) is computed as Ax, where A is an m x n matrix.

Linear Transformation

A transformation T that satisfies two properties: (i) T(u+v) = T(u) + T(v) for all u, v in the domain of T; and (ii) T(cu) = cT(u) for all scalars c and all u in the domain of T.

Superposition Principle

A property of linear transformations stating that T(c₁v₁ + c₂v₂ + … +cₚvₚ) = c₁T(v₁) + c₂T(v₂) + … + cₚT(vₚ) for any vectors v₁, …, vₚ and scalars c₁, …, cₚ.

Contraction (Linear Transformation)

A linear transformation T: R² → R² defined by T(x) = rx, where 0 < r < 1.

Dilation (Linear Transformation)

A linear transformation T: R² → R² defined by T(x) = rx, where r > 1.

Projection Transformation

A matrix transformation that projects points from a higher dimension onto a lower-dimensional plane (e.g., from R³ onto the x₁x₂-plane).

Shear Transformation

A matrix transformation that deforms a geometric shape (e.g., a square) into a sheared parallelogram, appearing as if one side is pushed while the base is fixed.