Ch1. Linear Algebra Review

Question 1

Norm 定義

Answer 1

1. 長度大於等於 0
2. ║ax║ = |a| ║x║
3. 滿足三角不等式 (║ x + y║ ≤ ║x║ + ║y║)

Norm 是 “向量的長度”，有多種定義方法，唯一要求是符合以上三個條件

Question 2

Lp-Norm

Answer 2

║x║p = 所有向量p平方相加，再開p次方根(幾何平均)

Question 2

L0-Norm

Answer 2

算向量中多少不為0的元素(不是Norm! 不合第二個條件)

Question 2

L2-Norm (Euclidean Norm)

Answer 2

║x║₂ = √(x₁² + x₂² + … + xₙ²)

Question 2

L1-Norm

Answer 2

║x║₁ = |x₁| + |x₂| + … + |xₙ|

Question 3

L∞-Norm

Answer 3

所有向量中最大的成員

Question 4

Cauchy-Schwarz Inequality

Answer 4

Let x ∈ Rn and y ∈ Rn then
|x T y| ≤ ║x║₂ ║y║₂
|x T y| 內積
重要性: 不管任何維度都適用

Question 5

Cosine Similarity

Answer 5

|x T y| / ║x║₂ ║y║₂
cosine 值越大(越接近1) -> 兩向量夾角越接近0度 -> 向量越接近

Question 5

Linearly independent

Answer 5

線性獨立
向量集合中，沒有一個向量能夠被其他向量的線性組合取代
(每個向量都無法被其向量表示出來)
—
Definition 1.2:A set S = {v1 ,…, vr} , where r ≥ 2 is

a) linearly independent iff not exist a vi ∈ S being a
linear combination of other vectors in S ;

b) linearly dependent iff exist a vi ∈ S being a linear
combination of other vectors in S ;

If S has only one element, linearly independent

Question 6

Basis

Answer 6

在線性代數中，基（英文：basis，又稱基底) 是向量空間裡某一群特殊的向量（稱為基向量），使得向量空間中的任意向量，都可以唯一地表示成基向量的線性組合（或線性組合的極限）。

If V is any vector space and S = {v1, v2, . . vn} is
a set of vectors in a finite-dimensional vector
space V, then S is called basis for V if :
a) S spans V
b) S is linearly independent (no redundant vectors)

finite-dimensional vector space if it is spanned by a finite set of vectors! (basis 是有限多的!)

Question 6

線性獨立在 Ax = 0 的特性

Answer 6

S = {v1 , v2 ,…, vr} is linearly independent if and only if
the only coefficients satisfying
x1v1 + x2v2 +…+ xrvr = 0 are x1 = x2 =… = xr =0

Ax = 0 only has trivial sol.!

Question 6

Dimension Theorem

Answer 6

Given a matrix A : m×n
Rank(A) + Nullity (A) = n
Rank(AT) + Nullity (A) = m

Question 6

Rank & Nullity

Answer 6

Given a matrix A : m×n
 Rank(A) = dim(Row(A)) = dim(Col(A))
 Nullity(A) = dim(Null(A))
 Rank(A) = Rank(AT) ; Rank(A) ≦ min(m , n)

Rank: 一個矩陣需要多少 row/col. 就夠了 (不需要原始大小!)

Question 7

row/column space

Answer 7

矩陣的 row/column vector 所 span 出的 vector space

Question 7

Eigenvalue & eigenvector

Answer 7

Given a square matrix A ∈ Rn×n, vector u ∈ Rn
(with u ≠ 0) is called the eigenvector of A if
Au = λx , for some λ ∈ R.

The scalar λ is called the eigenvalue
corresponding with u

Question 8

null space

Answer 8

滿足所有 Ax = 0 的解答所 span 出的 space