## Linear Algebra: Interactive Exercises

### Linear Combinations of Vectors

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

### Span of Vectors

- Visualize span of one vector in two dimensions
- Visualize span of two vectors in three dimensions
- Given vectors in the span of a vector u, find the multiples of u that produce the given vectors
- Given a multiple of a vector u, locate its position in the span of u
- Given vectors in the span of two vectors u and v, find the linear combinations that produce the given vectors
- Given linear combinations of vectors u and v, locate their positions in the span of u and v

### Linear Independence

- Determine if the set of given vectors is linearly independent

### Linear Transformations

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

### Projections and Orthogonal Complements

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