Linear Algebra with Applications: Pearson International Edition
This is a short but complete summary of the subject Maths 2 (lineair algebra) of the course Econometrics (AEO) of the University of Amsterdam (UvA). It covers everything you need to know in a brief but detailed manner using sophisticatedly generated word formulas. Achieved grade: 8,5
The length of a vector 𝑢 ⃗ ∈ ℝ𝑛 is defined by ‖𝑢 ⃗ ‖ = √𝑢
⃗ ∙𝑢⃗ = √𝑎2 + 𝑏 2 . In ℝ𝑛 there are n
unit vectors 𝑒1 , 𝑒2 , … , 𝑒𝑛−1 , 𝑒𝑛 with length 1. Take an arbitrary vector 𝑣 ≠ ⃗0, then the vector
1
𝑢
⃗ = ‖𝑣⃗‖ 𝑣 has length 1.
Vector 𝑣 is a linear combination of the vectors 𝑣1 , 𝑣2 , … , 𝑣𝑘 , if there are scalars
𝑐1 , 𝑐2 , … , 𝑐𝑘 ∈ ℝ such that: 𝑣 = 𝑐1 𝑣1 , 𝑐2 𝑣2 , … , 𝑐𝑘 𝑣𝑘 .
Summary: Normal form Parametric equation / Point normal equation
vector representation
Line in ℝ2 𝑎𝑥 + 𝑏𝑦 = 𝑐 𝑥 = 𝑝 + 𝑡𝑑 𝑛⃗ ∙ (𝑥 − 𝑝) = 0
Line in ℝ3 𝑎 𝑥 + 𝑏1 𝑦 + 𝑐1 𝑧 = 𝑑1 𝑥 = 𝑝 + 𝑡𝑑 𝑛 ∙ (𝑥
⃗⃗⃗⃗ ⃗⃗⃗1 − 𝑝⃗⃗⃗1 ) = 0
{ 1 { 1
𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 𝑧 = 𝑑2 𝑛2 ∙ (𝑥
⃗⃗⃗⃗ ⃗⃗⃗⃗2 − 𝑝
⃗⃗⃗⃗2 ) = 0
3
Plane in ℝ 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑥 = 𝑝 + 𝑡𝑑 + 𝑠𝑢 ⃗ 𝑛⃗ ∙ (𝑥 − 𝑝) = 0
𝑎
𝑎
Normal of line in ℝ2 = ⃗⃗⃗⃗
𝑁𝑙 = ( ) = 𝑛⃗, normal of line/plane in ℝ3 = 𝑛⃗ = (𝑏)
𝑏
𝑐
Direction vector = 𝑑 , normal is orthogonal to the direction vector (cylinder - Line in ℝ3 )!
1
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