Linear Algebra

Study of vectors, matrices, and linear transformations.

Vectors and Vector SpacesMatrices and Matrix OperationsSystems of Linear Equations
Determinants and InversesEigenvalues and EigenvectorsLinear Transformations and Their Properties
Computer Graphics and AnimationData Science and Machine LearningEngineering and Physics Applications
Problem-Solving with MatricesVisualizing Vectors and TransformationsBreaking Down Word Problems
Vectors and Vector S...Matrices and Matrix ...Systems of Linear Eq...Determinants and Inv...Eigenvalues and Eige...Linear Transformatio...Computer Graphics an...Data Science and Mac...Engineering and Phys...Problem-Solving with...Visualizing Vectors ...Breaking Down Word P...
Advanced Topics

Linear Transformations and Their Properties

What Is a Linear Transformation?

A linear transformation is a rule that takes vectors and maps them to new vectors, following two rules:

  1. Adding vectors before or after mapping gives the same result.
  2. Scaling a vector before or after mapping gives the same result.

Matrix Representation

Every linear transformation can be represented by a matrix! So, multiplying a vector by a matrix is the same as applying a linear transformation.

Understanding Properties

  • Kernel: All vectors that get sent to zero.
  • Image: All possible outputs of the transformation.

Why Care About Linear Transformations?

  • They let us rotate, scale, or flip objects in graphics.
  • They help us understand how data changes shape.
  • They're used in robotics, engineering, and more.

Real-World Examples

  • Rotating an image in a video game.
  • Scaling a 3D model up or down.

Examples

  • Rotating a shape using a transformation matrix.

  • Finding the kernel of a transformation to see which vectors map to zero.

In a Nutshell

Linear transformations use matrices to move, stretch, rotate, or flip vectors while preserving structure.

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Computer Graphics and Animation