Linear Algebra: Interactive Exercises

Linear Combinations of Vectors

  1. Compute and visualize linear combinations of two vectors:
  2. Visualize a given linear combination of two vectors and compute the coefficients of the two vectors:

Span of Vectors

  1. Visualize span of one vector in two dimensions
  2. Visualize span of two vectors in three dimensions
  3. Given vectors in the span of a vector u, find the multiples of u that produce the given vectors
  4. Given a multiple of a vector u, locate its position in the span of u
  5. Given vectors in the span of two vectors u and v, find the linear combinations that produce the given vectors
  6. Given linear combinations of vectors u and v, locate their positions in the span of u and v

Linear Independence

  1. Determine if the set of given vectors is linearly independent

Linear Transformations

  1. Given a linear transformation, visualize outputs for various inputs
  2. For a linear transformation, pick various inputs and see how the outputs vary
  3. Visualize the addition condition in the definition of a linear transformation
  4. Visualize the scalar multiplication condition in the definition of a linear transformation

Projections and Orthogonal Complements

  1. Visualize orthogonal projections of various vectors onto a vector u
  2. Visualize the orthogonal complement of a two dimensional subspace in a three dimensional vector space