The Four Subspaces of a Matrix
The " Four Subspaces " - Column Correspondence Property
1)Matrix A
2) Matrix A ( rref. ) - Labeled number - 1 -
3)Matrix A (rref.) - Transpose - Labeled number - 1a -( A transpose (rref) or Left Null Space.)
4) Number - 1b :" Left Null Space of " A - rref. - Transpose ": Answer; Solutions of A v = 0 - 1b comes from 1a.
5)Number 2 -: " Column Space of A " : The Pivot Columns ; - pc (Blue) with Red arrows
6)Number 3 : " Row Space of " A (rref.) " : The Pivot Rows ; - PR (Red) - All Combinations of the Pivot Rows- Were I have the equals sign crossed out, I mean; Just that the equations were not moved to the other-side of zero- making a sign change.
7)Number 4 : " Null Space of A (rref.) " : The Green Arrow Columns - Solutions to A x = 0
8) Number 4A : The Null Space Solutions in Form.
9) Notice at the bottom right of each of the four subspace Matrix ; the designations, ; Pr, Pc, NS, and m with their corresponding rows and column numbers. Those small designations puts this into perspective.
I.E. : Pivot Rows, Pivot Columns, Null Space, and Row " m " with their column and row numbers; what ever applies...