Equivalent System: two systems with the exact same set of solutions. Valid Operation: lead to Equivalent System (each operation connects two Equivalent System in reduction)
Multiply non-zero constants
Add multiples of a row to another
Linear Transform
Triangular System:
Example: \begin{cases} 2u + v + w = 5 \\ -8v -2w = -12 \\ 1w = 2 \\ \end{cases}
How to get to Triangular System:
Singular System: If A is singular, the linear system Ax = b has either no solution or infinitely many solutions. Non-singular System: it has exactly one solution. Translation: P_a is a translate of P_2 if there is a vector v such that adding v to the parts of P_n, we get position of P_2. Inconsistent: no solution, there exists a contradiction
Row Picture: draw lines Column Picture: draw vectors without solution
Solution | Row-Picture | Column Picture |
---|---|---|
1 | 3 plane a point | 3 vector not in plane |
\inf | all planes coincide | all 4 vectors in line |
\inf | three planes form line | all 4 vectors in plane |
0 | two parallel planes, form 2 lines | 3 vectors in plane |
0 | form 3 lines | 3 vectors in line |
// WARNING: above is too naive and questionable
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