Row Picture and Column Picture of Equation
01. Two Dimensions Case
Equation
Matrix Format
Row Picture
In the view of row picture, each row of equation is a line in 2-D space. And the solution of the equation system is the intersection of the two lines.
Column Picture
In the view of column picture, each column of A is a vector in 2-D space. And the solution of the equation system is the weight of vectors' linear combination that points to b.
02. Three Dimensions Case
Equation System
Matrix Form
Row Picture
In the view of 3D-row picture, each row of the equation system is a plane in 3D space, and the solution point is the intersection of the three points.
Column Picture
In the view of 3D-column picture, each column of A is a vector in 3D space, and the solution is the weight of vectors' linear combination that points to b.
03. About Solvability
The two statements below are equivalent.
1. Can I solve Ax = b for every b?
2. Do the linear combinations of A's columns fill the whole 3D space.
