column space

column space is the space spanned by the columns of matrix . It’s a subspace of vector space

The dimension of the column space equals the rank of

Relationship

dimension of column space = the number of vectors in the basis of column space = number of linearly independent columns in matrix A = rank A

A sister term is row space

Relationship with linear system

Relationship with fundamental subspaces

Consider the linear system

The system is consistent system if and only if we can find the solution such that

The left-hand side forms the column space. So the solution only exists if is in the column space .

Approach 1 with column space

Strategy to determine if the system (where is consistent

1. Find the basis for column space
2. Determine the dimension of this column space & dimension of
3. If , the column space spans and thus contains . The system is consistent

Approach 2 with left null space

Strategy to determine if the system (where has no solution

Check if is orthogonal to the left nullspace

1. transpose
2. Reduce to rref and find the parametric solution
3. Find the basis for left nullspace
4. Compute the dot product of every vector in the column space of with every vector in the basis above. If all results are zero, the linear system has no solutions.

Examples

Link to original

Relationship with linear transformation

Transclude of linear-map#connection-with-column-space