Linear transformations

Understanding Linear Transformations

  • A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
  • This can be described using matrix multiplication where the matrix represents the linear transformation.
  • Graphically, linear transformations can be visualised as changing shapes while preserving points in line and their proportions.
  • The standard matrix of a linear transformation is the matrix for the transformation relative to the basis vectors.

Properties of Linear Transformations

  • The additive property says that the transformation of the sum of two vectors equals the sum of the transformations of the vectors.
  • The homogeneity property says that the transformation of a scalar multiplied by a vector equals the scalar multiplied by the transformation of the vector.
  • A transformation is linear if it satisfies both the additive and homogeneity properties.
  • A transformation is affine if it is linear plus a vector.

Types of Linear Transformations

  • Translation is a linear transformation that moves every point of a figure or space by the same amount in a given direction.
  • Scaling is a linear transformation that enlarges or shrinks objects by a scale factor.
  • Rotation is a transformation that rotates points in a plane or space around a fixed center of rotation.
  • Shear is a transformation that displaces each point in the direction parallel to a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin.
  • Reflection is a transformation that flips a figure over a line of reflection.

Eigenvalues and Eigenvectors

  • In a linear transformation, an eigenvector is a nonzero vector that only changes by a scalar factor when the transformation is applied to it.
  • The scalar factor is the eigenvalue associated with the eigenvector.
  • The process of finding eigenvectors and eigenvalues is called eigendecomposition. This is particularly useful for understanding the properties of the linear transformation, such as whether it has an inverse, and if so, want that inverse is.

Homogeneous Systems of Equations

  • Linear transformations often result in systems of linear equations to solve.
  • A system of linear equations is homogeneous if all terms on the right hand side are zero. This corresponds to a linear transformation that maps to the zero vector.
  • Homogeneous systems always have at least one solution, namely, the trivial solution where all variables equal zero.
  • The methods to solve such systems include gaussian elimination, the method of undetermined coefficients and determinants and cofactors.