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Linear transformation

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Linear transformations are a special type of function that maps vectors in a vector space to another vector while preserving vector addition and scalar multiplication operations. Examples include scaling, rotating, and projecting vectors in different dimensions. Linear transformations are described using rules like matrix multiplication and maintain the properties of vector addition and scalar multiplication.

Linear Transformation#

Linear transformation is a special type of function that maps vectors in a vector space to another vector while preserving vector addition and scalar multiplication operations.

Basic Properties of Linear Transformation#

A linear transformation T satisfies the following two conditions:

  • Additive Preservation: T(u+v)=T(u)+T(v)
  • Scalar Multiplication Preservation: T(cv)=cT(v)

Examples of Linear Transformation#

Example 1: Scaling Transformation#

Consider a function T that maps any vector v=(x,y) in two-dimensional space to a new vector T(v)=(2x,2y). This function is a linear transformation because it satisfies the preservation of addition and scalar multiplication. Geometrically, this transformation symmetrically enlarges all points on the plane about the origin by a factor of two, i.e., scaling each point.

Example 2: Rotation Transformation#

In two-dimensional space, a vector can be rotated around the origin by multiplying it by a specific matrix. The linear transformation that rotates a vector v counterclockwise by an angle θ around the origin can be represented as:

T(v)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)(xy)T(\mathbf{v}) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

This transformation is linear because it preserves vector addition and scalar multiplication operations.

Example 3: Projection Transformation#

A projection transformation projects a vector v=(x,y,z) in three-dimensional space onto the XY plane, which can be represented as:

T(v)=(x,y,0)T(\mathbf{v}) = (x, y, 0)

This transformation ignores the z-coordinate of each point, projecting the points onto the XY plane, and is also a linear transformation.

Key Concept#

Linear Transformation

Linear transformation is a method used in mathematics to describe mappings between vector spaces. It transfers a vector from one space to another while maintaining the properties of vector addition and scalar multiplication through a set of rules (such as matrix multiplication).

Related Knowledge or Questions
[1] [[How are linear transformations composed using matrices?]]
[2] [[How to represent reflection transformations in three-dimensional space using matrices?]]
[3] [[How are matrix representations of linear transformations applied in image processing?]]

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