Vector and Matrix notable facts

Gauss Jordan Elimination

• Each operation is equivalent to multiply the augmented matrix to a fundamental matrix ($E_1$) or something
• Because of the properties of matrix multiplication, we can find the inverse applying GJ to AI = B and getting IA-1 = D
• If we get a 0 in the left part of the augmented matrix after GJ, the matrix is not invertible

Determinant is defined as a function

The definition starts from the identity matrix and applies linearity across columns. Determinants have the following key properties:

• Inverting a row changes the sign of the determinant
• …
• …

Questions: — Cramer rules on solving a linear system — How does that impact linear combinations? Can we show from cramer rules that if the determinant is zero, a certain number of vectors are not linearly independent

What is the scalar triple product?

Inner and cross product

• Inner product is equal to v1 * v2 * cos(a) because… why?
• Cross product is a vector obtained by putting a row e1,e2,e3 … en and computing the determinant.
  • Its norm is equal to v * x * sin(0)
    • This come from a square property
  • Other properties:
    • associativitiy
    • multiply by a factor
    • null
  • in 2d, the vector is just vertical outside the 2d plan

See notebook for an interactive visual proof that |a x b| equals the parallelogram area.

Line and plane geometry

Line form:

• Slope
• Intercept x/a + y/b = 1
• Normal ax + by +c = 0
• Parametric ? what is exactly? Plane:
• Intercept x/a + y/b + z/c = 1
• Normal (Ax + By +Cz + D) = 0

What are the key Idea about generating plane and lines from vector?