Linear Algebra/A

Available as a playlist https://www.youtube.com/playlist?list=PL01A21B9E302D50C1 (1-43)

Introduction to Linear Algebra
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Definition
Linear Algebra is Linear Algebraic Geometry
 * (Affine) Geometry of n-dimension space + linear transformations
 * Applications: engineering, economics, physics, graphics, computer science, mathematics, statistics, finance, robotics, chemistry

Affine Geometry
2D
 * geometry of parallels, but not perpendicular

Grid Plane
We can describe relative distance.
 * No distance measurements
 * No special point (origin)
 * No angle measurement
 * Vectors (x,y)

Subpoints

 * We can divide any segment into n equal pieces
 * Vectors can have rational coefficients
 * Rational numbers are a/b, where a and b are integeres, b!=0. a/b=c/d iff ad-bc=0

Vector Arithmetic

 * Scalar multiplication
 * Addition of vectors

Two Affine Planes

 * Comparing two different basis vectors
 * Converting between basis systems

Main Problem I of Linear Algebra

 * Write one basis vector system in terms of another
 * 2x2 determinant is needed

Questions
Find x_1,x_2,x_3 in terms of y_1,y_2,y_3
 * What happens if ad-bc = 0?
 * What is the geometric meaning of ad-bc?
 * How do we generalize to 3D?