subspaces

Summary

Properties of a subspace

system must be consistent, for its solution set to be a subspace

Solution set of homogeneous systems

• solution space/is a subspace

Checking if a vector is in a subspace

• check if the corresponding x,y,z values correspond to

Concept

Solution set to a linear system

solution set is not neccesarily a subspace

Solution set of an inconsistent system

Subspace is a span

Affine spaces

• similar to relation between homogeneous and non-homogeneous systems
• its displacement is the particular solution

visualise a plane that doesn’t pass through the origin

Application

Converting between implicit and explicit forms

Proving that a span is equal to a subspace

note: the subspace defines a plane, which is defined by 2 basis vectors, therefore S is not linearly independent

Affine spaces