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.

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。