Linear Algebra

Study of vectors, matrices, and linear transformations.

Vectors and Their OperationsMatrices and Matrix ArithmeticLinear Transformations
Determinants and InversesEigenvalues and EigenvectorsVector Spaces and Subspaces
Computer Graphics and AnimationData Science and Machine LearningRobotics and Engineering
Breaking Down Word ProblemsChecking Your Solutions with InversesVisualizing Problems to Find Solutions
Vectors and Their Op...Matrices and Matrix ...Linear Transformatio...Determinants and Inv...Eigenvalues and Eige...Vector Spaces and Su...Computer Graphics an...Data Science and Mac...Robotics and Enginee...Breaking Down Word P...Checking Your Soluti...Visualizing Problems...
Advanced Topics

Vector Spaces and Subspaces

The World of Vector Spaces

A vector space is a collection of vectors that you can add together and scale, and the result will always be another vector in the same space.

  • Subspaces are smaller "rooms" inside the bigger vector space where the same rules apply.

Why Care?

Understanding vector spaces helps us describe solutions to equations, create computer graphics, and more.

Examples of Subspaces

  • All vectors in the plane \( (x, y) \) form a vector space.
  • All vectors along a line through the origin are a subspace.

Try This

  • Check if the set of all vectors \( (x, 0) \) is a subspace.
  • Find if a set of polynomials forms a vector space.

Examples

  • All solutions to a homogeneous system form a subspace.

  • The set of all 2D vectors is a vector space.

In a Nutshell

Vector spaces are sets where you can add and scale vectors, and subspaces are smaller sets with the same rules.

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Computer Graphics and Animation