Udacity Data Scientist Nanodegree : Prerequisite — Linear Algebra

Lesson 1: Introduction / Lesson 2: Vectors / Lesson 3: Linear Combination / Lesson 4: Linear Transformation and Matrices

Joe Chao
4 min readMar 23, 2021

Essence of Linear Algebra

  • Geometric intuition is helpful for what tools to use in this problem.
  • Numerical understanding can make you actually compute those computations.

Vectors, what even are they? Part 1

  • Physic perspective: Vectors are arrow pointing in space. What defines a given vector, is its length and the direction it’s pointing. With the same factors(length and direction), you can move it all around and it‘s the same vector.
  • CS perspective: Vectors are ordered list of numbers.
  • Mathematic perspective: Vector can be anything where there’s sensible notion of adding two vectors and multiplying a vector by a number.
  • Vector definition in this video: An arrow in a coordinate system with its tail sitting at the origin.

Vectors, what even are they? Part 2

  • Each element in the vector, also called component or coordinate.
  • 向量的座標說明了向量如何從位於原點的尾部出發,到達終點所在的頭部。
  • 為了跟點做區分,通常我們會用矩陣形式去寫他的座標位置。
  • 三維空間:[x, y, z]^T

Vectors, what even are they? Part 3

  • 向量加法:移動第二個向量使其尾部落在第一個向量的頭部上。從第一個向量的尾部畫到第二個向量的頭部,畫一個新向量,就是他們相加的結果。
  • 向量乘法:向量的係數(Scalars)是把它拉長/縮短幾倍。負數係數則是會先翻轉這個向量,然後變成原本的係數倍(這個行為 — Scaling)

Vectors- Mathematical Definition

What is a Vector? The plain explanation would be that a vector is an ordered list of numbers.

n Dimensional Vector

As in the video let’s put this into a more visual context and focus on a 2D vector of the field of real numbers. In other words, we will focus on a vector in R^2, which defines all points on the plane.

Vector Transpose

Magnitude and Direction

  • Each vector holds the magnitude as well as the direction of the movement.
  • The symbol we use for the magnitude(向量的大小) is || ||.
  • Direction: Radian or degree.

Linear Combinations, Span and Basis Vectors. Part 1

  • 如果你有一對數字可以表示一個向量,可以把各座標分別想成一個純量。
  • 有幾個比較特殊的向量(i-hat = unit vector,x軸的單位向量)跟 j-hat,y軸的單位向量。可以把一般的向量想成是 i-hat 跟 j-hat 的幾倍 — 線性組合。i-hat and j-hat are the “basis vectors”. 透過這些 basic vector 的轉換,可以得到所有二維平面上的座標點。
  • 但如果兩向量共線了,就無法得到全部的座標。

Linear Combinations, Span and Basis Vectors. Part 2

  • 如果要研究一個向量,通常會畫箭頭,如果要研究向量集合,通常會把向量簡化成一個點,頭部是點,而尾部依舊是原點。
  • The Span of those vectors (sometimes also referred to as the Linear Span) is the set of all possible linear combinations of those vectors.
  • 二維:多數向量組合的 span 是二維空間,但若向量共線,則為一條直線。
  • 三維:兩個向量的Span — 一個平面。三個向量的 Span — 三維空間、一個平面(若第三個向量落在前兩個向量的平面上)。
  • 線性相關:只要你有多個向量,且可以在不減少 Span 的情況下刪除其中一個向量,就可以說這些向量是線性相關的。反之則稱為 Linear Indepedent。
  • When one vector can be defined as a linear combination of the other vectors, they are a set of linearly dependent vectors. — 線性相關的方程式有無限多組解。

Linear Transformation and Matrices . Part 1

  • 投入 input 然後輸出 output。為什麼我們要說變換而不是函數呢?因為變換更貼切。
  • 有兩個特點:所有線都必須是直線、原點必須固定不動。網格線必須平行且等距分佈。比如說旋轉。

Linear Transformation and Matrices. Part 2

  • 只要紀錄 i-hat 跟 j-hat 的移動方向,就可以知道整個空間的變動。

Linear Transformation and Matrices. Part 3

  • Shear: i-hat 不變 j-hat 到了(1, 1)
  • sum up: 可以把矩陣看成空間向量的組合。



Joe Chao