Lecture 002

Gaussian Elimination

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)

Triangular System:

visualizing 3x3 system

visualizing 3x3 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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