Row Picture and Column Picture of Equation

01. Two Dimensions Case

Equation

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Matrix Format

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Row Picture

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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

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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

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Matrix Form

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Row Picture

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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

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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.