linear combinations

Summary

Properties of linear span

Algorithms

Relationships between spans

to prove euality, prove subset in both ways

Concept

Linear combinations are vector equations/linear systems

Linear span

• set of all linear combinations of the vectors

each linearly independent vector in the span represents one dimension of freedom

Check if vector is in a span

Check if linear combination spans n-space

the span must have enough dimensions of freedom to reach every vector

Geometric representation of span as lines in R³ and planes in R³

represents the point of higher dimensional space as the number of non-parallel vectors are included
dimension is equal to the number of linearly independent vectors in the span

Application

Solving a linear combination

combine the column vectors into a matrix and solve for the unknown coefficients

Finding vectors ouside of the span