dimensions

Summary

Relations between subspaces

Dimension of a solution space

• number of non-pivot columns

Equivalent checks for basis

in essence to check do

Concept

Dimension of a subspace

• minimum number of vectors required to span a subspace -> number of vectors in any basis of the space
• number of degress of freedom(linearly independent vectors) in the subspace

every basis of a subspace has the same number of vectors

Spanning set theorem

if a set spans , then either it or its subset is a basis for

Linear independence theorem

a linearly independent subset of , is either a basis for or is part of a superset that is a basis for

Check for basis with dimensions

number of basis vectors match the dimension and in or spans

Application

Check for basis

Subspace implicit <-> explicit form via coefficient matrix