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...
Here is one 4 x 6 Matrix broken down into it's parts - The four Subspaces. But , also - the Column Correspondence principle is in play as well. The pivot columns, when added together or multiplied- can be used to obtain the vector(s) to( it's, or their) Right- which are the Dependent Vectors in the Columns of X2, X5, X6. The Columns of X1, X3, X4 are Linearly Independent and make up the Pivot Columns and Pivot Rows, respectively and compliment each other. i.e. Pivot Columns and Pivot Rows, which are independent.