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)

• 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:

  • 1, 2 -> 2
  • 1, 3 -> 3
  • 2, 3 -> 3
  • done!

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

• might not be solution if three vectors are in the plane (3 vector, at most 2 direction)

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