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# 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.

# 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 || ||.

# 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。