🍊 Linear Algebra

🍋 Matrices, Vectors, Linear Transformations

• 🍋 “Solve the matrix,” transformations in 2D/3D, cryptography ciphers.

🍋‍🟩 Eigenvalues & Eigenvectors

• 🍋‍🟩“Population growth”, or Markov chain re-interpreted.

🥝 Linear Systems

Linearity: When Stuff Scales Nicely

A system is linear if addition and multiplication work predictably. That means:

• If you sum two inputs, is like summing the outputs.
• If you scale an input, the output scales by the same factor.

graph LR
    x@{ shape: dbl-circ, label: "𝓍" }
    x --> fx
    subgraph LinearSystem
        style LinearSystem fill:darkred,stroke:tomato,stroke-width:2px
        fx@{ shape: hex, label: "𝒻(𝓍)" }
end

graph LR
    x@{ shape: dbl-circ, label: "𝓍" }
    y@{ shape: dbl-circ, label: "𝓎" }
    xy@{ shape: circ, label: "𝓍+𝓎" }
    a@{ shape: dbl-circ, label: "➕"}
    x --o a
    y --o a
    a --> xy
    xy --> fxy
    x --> fx
    y --> fy

    style a fill:silver ,stroke:#333,stroke-width:4px

    subgraph Adding
        style Adding fill:darkred,stroke:tomato,stroke-width:2px
        fx@{ shape: hex, label: "𝒻(𝓍)" }
        fy@{ shape: hex, label: "𝒻(𝓎)" }
        fxy@{ shape: hex, label: "𝒻(𝓍+𝓎)" }
        fxay@{ shape: hex, label: "𝒻(𝓍)+𝒻(𝓎)" }
        fa@{ shape: dbl-circ, label: "➕"}
        fxy <==> fxay
        fx --o fa
        fy --o fa 
        fa --> fxay
    end
    style fa fill:silver ,stroke:#333,stroke-width:4px

graph LR
    x@{ shape: dbl-circ, label: "𝚊𝓍" }
    x --> fx
    subgraph LinearSystem
        style LinearSystem fill:darkred,stroke:tomato,stroke-width:2px
        fx@{ shape: hex, label: "𝒻(𝚊𝓍)" }
        afx@{ shape: hex, label: "𝚊𝒻(𝓍)" }
        fx <==> afx
     end

For example, if you buy 2 apples for $1 each, the total cost is simply:

2×1=2

And if you buy 3 apples instead, the cost is:

3×1=3

Nothing unexpected happens: the relationship is straightforward. This makes linear functions incredibly useful in engineering, physics, and even art (perspective drawing relies on linear transformations!).

---

Conjuntos:

• 𝑎∈𝔸 𝑎 Pertenece a 𝔸
• ℕ Naturales
• ℤ Enteros
• ℚ Fraccionales
• ℝ Reales
• ℂ Complejos
• 𝔸×𝔹 Binómio: ｛(𝑎,𝑏) | 𝑎∈𝔸, 𝑏∈𝔹｝
• ℝ²=ℝ×ℝ``𝑣∊ℝ²: {𝑣 := (𝑥,𝑦); 𝑥,𝑦∊ℝ}
• Circunferencia: 𝕊² | 𝑣 := (𝑥,𝑦); 𝑥²+𝑦²=𝟏
• Esfera: 𝕊³ | 𝑣 := (𝑥,𝑦,𝑧); 𝑥²+𝑦²+𝑧²=𝟏
• Cilíndro: ℂ³ | 𝑣 := (𝑥,𝑦,𝑧); 𝑥²+𝑦²=𝟏

Polinomios

ℙⁿ(ℝ) = { 𝑎₀+𝑎₁+𝑎₂+𝑎₃.+.𝑎ₙ | 𝑎ᵢ∊ℝ }

Operaciones de conjuntos

• Unión 𝔸∪𝔹 : {𝑥 | 𝑥∈𝔸 ⋁ 𝑥∈𝔹}
• Intersection 𝔸∩𝔹 : {𝑥 | 𝑥∈𝔸 ⋀ 𝑥∈𝔹}
• Complement ∁𝔸: {𝑥 | ￢𝑥∈𝔸 }
• Relative Complement 𝔹∖𝔸: {𝑥 | ￢𝑥∈𝔸 ⋀ 𝑥∈𝔹}

Greatest common divisor

if the gcd of (x,y) =1
then x and y are coprimes, or relative primes.

$gcd (12, 15) = 3$

Euclidean:

$n = qd + r$
$gcd(n,d) = gcd(d,r)$

Least common multiple

The smallest integer that divides by both

$if$ $x ∣ n ∧ y ∣ n$
$⇒ lcm(x,y) ∣ n$

$d ∣ x, y$
$⇒ d ∣ gcd(x,y)$

Greatest Common divisor: Jars 🏺

You have two jars
One holds 3 litres
one holds 5

Is it possible to measure out 1 later of water?
Generally, what quantities is it possible to measure?

2(3)🏺 - 1(5)🏺 = 1 💧

What can we do with the two jars =

$a·3 + b·5$

negatives is removing

$n( 2(3) - 5 ) = 3·2n -5·n = n$

We can repeat the process of adding two jars of 3L and extracting one of 5L.

In general

Given integers x and y, determine all possible values of the form

${ ax + by : a, b ∈ ℤ }$

Example

Consider two jars of 4L and 10L.

$4a + 10b$

We can only get even numbers.
proof:

$4a + 10b = 2$

or for any even number

$m(4a + 10b) = 4(am) + 10(bm) = 2m$

for

$a = -2$
$b = 1$

we see we can get any even number.

In general $ax + by$ is a linear combination.

The key is to know the smallest possible combination.

The greatest common divisor of two integers x and y in the smallest positive integer that is a linear combination of x and y

the set of linear combinations of x and y are the multiples of gcd(x,y)

${ ax + by : a, b ∈ ℤ }$
$=$
${ 𝑚 · gcd(x,y) : 𝑚 ∈ ℤ }$

As we have already seen, the answer is yes, we can measure 1 liter given 3-liter and 5-liter jars. Now we know the reason this is possible is that $gcd(3, 5) = 1$.

Linear Algebra

Vector Algebra

Addition

V⃗ + U⃗ = W⃗

∀i ( w[i] = v[i] + u[i] )

Multiplication

W⃗ = 𝑎V⃗

∀i ( w[i] = 𝑎·v[i] )

Linear Combination

Addition and multiplication combined

𝑎V⃗ + 𝑏U⃗ = W⃗

∀i ( w [i] = 𝑎·v[i] + 𝑏·u[i] )
𝑎,𝑏∈ℝ

For any a and b of two vectors they can either fill a space or form a line

Vector Product

V⃗·U⃗ = 𝚠

Length

||V⃗|| = √V⃗·V⃗

Zero Vector

０⃗: ０[i] = 0

０⃗ is into every linear combination (𝑎,𝑏 = 0)

Linear components

𝑎V⃗ : makes a line
𝑎V⃗ + 𝑏U⃗ : Makes a plane
𝑎V⃗ + 𝑏U⃗ + 𝑐W⃗ : Makes a volume
etc

Ordinary Equations

A set of ordinary equations can be represented by a linear combination of vectors, or a Matrix Equation.

 2𝑎  －  𝑏 = 1
- 𝑎  ＋ 2𝑏 = 0
═════════════════════════════
𝑎 (2, -1) + 𝑏 (-1,2) = (1,0)

════════════════════
⎡  2  -1⎤⎛𝑎⎞ = ⎡ 1 ⎤
⎣ -1   2⎦⎝𝑏⎠ = ⎣ 0 ⎦

Dot Product

V⃗ ⋅ U⃗ = 𝐰
𝐰 = ∑ 𝐯ᵢ×𝒖ᵢ

Length

||V⃗|| = √ V⃗⋅V⃗ ]

Unit Vector

𝑉̂ = V⃗ ∕ |V⃗|
⠀⠀|| 𝑉̂ || = 1

In 2D

𝜃̂ = (cos𝜃, sin𝜃)

Perpendicularity

V⃗ ⋅ U⃗ = 0 ⟹ V⃗∟ U⃗
⠀⠀⠀⠀⠀⠀⠀⟹ ||V⃗||² ＋ ||U⃗||² = ||V⃗ － U⃗||²

V⃗ ⋅ U⃗̂ = cos 𝜃
⠀⠀⠀ 𝜃 := smallest angle ∡𝑉,𝑈

cos 𝜃 =
V⃗ ⋅ U⃗
────
||V⃗|| || U⃗||

Schwarz Inequality

|V⃗ ⋅ U⃗| ≤ |V⃗| × |U⃗|

Triangular Inequality

|V⃗ + U⃗| ≤ |V⃗| + |U⃗|

Matrix times Vector

⎡ a11  a12⎤⎛𝑎⎞ = ⎛𝑎·a11 ＋ a12·𝑏⎞
⎣ a21  a22⎦⎝𝑏⎠   ⎝𝑎·a21 ＋ a22·𝑏⎠

Matrix

: Vector of Vectors

A⃣ = A⃗⃗
══
⎡A⃗₁⎤
⎢A⃗₂⎥ 
⎢⢰ ⎥
⎣A⃗ₙ⎦
═════
⎡a₁₁ a₁₂ a₁₃ ⋯ a₁ₘ ⎤
⎢a₂₁ a₂₂ a₂₃ ⋯ a₂ₘ ⎥ 
⎢⢰    ⋮    ⋮  ⋱ ⋮  ⎥
⎣aₙ₁  aₙ₂  aₙ₃ ⋯ aₙₘ​⎦

Matrix × Vector

A⃣ v⃗
= (A⃗₁· v⃗ , A⃗₂· v⃗ , A⃗₃· v⃗ , ⋯ A⃗ₙ· v⃗ )

Inverse

𝐴⃣ v⃗ = b⃗
v⃗ = 𝐴⃣⁻¹ b⃗

𝐴⃣⁻¹ 𝐴⃣ = 𝐼⃣
𝐼⃣ = ( u⃗₁, u⃗₂, u⃗₃ … u⃗ₙ)
u⃗ᵢ = { 1 at i, 0 else)

Calculate Inverse

Transform 𝐴⃣ into 𝐼⃣ , then apply the same transformations into 𝐼⃣ . Thats the inverse.

⎡1  2⎤⎡1 0⎤ 
⎣3  4⎦⎣0 1⎦
═══════════════════ 2ª - 3·1ª
⎡1   2⎤⎡ 1 0⎤ 
⎣0  -2⎦⎣-3 1⎦
═══════════════════ 1ª + 2ª
⎡1   0⎤⎡-2 1⎤ 
⎣0  -2⎦⎣-3 1⎦
════════════════════ -2ª/2
⎡1  0⎤⎡ -2     1 ⎤ 
⎣0  1⎦⎣ 1.5  -0.5⎦

To Annulate a matrix with the inverse, apply matrices left to right (as presented)

Dependence

w⃗ is dependent if
w⃗ = 𝑎v⃗ + 𝑏u⃗

A matrix has independent column is invertible and the solution is unique.

Solving Linear Equations

𝐴⃣ x⃗ = b⃗
⎧ 𝑥₁ 𝑎₁₁ + ⋯ + 𝑥ₙ𝑎₁ₙ  = 𝑏₁
⎨ ⋮                    ⋮
⎩ 𝑥₁ 𝑎ₘ₁ + ⋯ + 𝑥ₙ𝑎ₘₙ   = 𝑏ₘ

-- Example
⎡1  -2⎤⎡𝑥⎤ = ⎛ 1 ⎞
⎣3   2⎦⎣𝑦⎦   ⎝ 11⎠

Matrix Transpose

B⃣ = A⃣ᵀ : { 𝑏ᵢⱼ = 𝑎ⱼᵢ}

Gaussian method

Column 1. Use the first equation to create zeros below the first pivot.
Column 2. Use the new equation 2 to create zeros below the second pivot.
Columns 3 to n. Keep going to find all n pivots and the upper triangular U.

1. A linear system A⃣ x⃗=b⃗ becomes upper triangular U⃣x⃗=c⃗ after elimination.
2. We subtract ℓᵢⱼ times equation j from equation i, to make the (i, j) entry zero. 
3. The multiplier is ℓᵢⱼ= entry to eliminate in row i : pivot in row j

• Pivots can not be zero!

4. When zero is in the pivot position, exchange rows if there is a nonzero below it.

5. The upper triangular Ux = c is solved by back substitution (starting at the bottom).
6. When breakdown is permanent, Ax = b has no solution or infinitely many.

Rules of Matrices

B⃣(A⃣ 𝑥⃗) = (B⃣A⃣) 𝑥⃗
A⃣B⃣ = C⃣  ⇏ B⃣A⃣ = C⃣

Augmented Matrix

[𝐴 | 𝚋]

Matrices do something

A⃣ 𝑥⃗= 𝑥₁ times column 1 + · · · + 𝑥ₙ times column n.
And (A⃣𝑥⃗)ᵢ = ∑ⱼ₌₁ 𝑎ᵢⱼ 𝑥ⱼ

Identity matrix = I⃣
elimination matrix = E⃣ᵢⱼ using ℓᵢⱼ
exchange matrix = P⃣ᵢⱼ.

Multiplying 𝐴⃣ x⃗ = b⃗ by E⃣₂₁ subtracts a multiple ℓ₂₁ of equation 1 from equation 2. The number -ℓ₂₁ is the (2, 1) entry of the elimination matrix E₂₁

For the augmented matrix [A⃣ b⃗], that elimination step gives [ E⃣₂₁A⃣ E⃣₂₁b⃗]

When A⃣ multiplies any matrix B⃣, it multiplies each column of B⃣ separately.

A⃣ₘₓₙ × B⃣ₙₓₚ = 
     ［ B͎̅₁    , B͎̅₂     , ⋯    B͎̅ₚ ] 
⎡A⃗₁;⎤ ⎡ A⃗₁·B͎̅₁ , A⃗₁·B͎̅₂ ,⋯, A⃗₁·B͎̅ₚ;⎤  
⎢A⃗₂;⎥ ⎢ A⃗₂·B͎̅₁ , A⃗₂·B͎̅₂ ,⋯, A⃗₂·B͎̅ₚ;⎥
⎢⋮  ;⎢⎥ ⋮                        |
⎣A⃗ₘ ⎦ ⎣A⃗ₘ·B͎̅₁  , A⃗ₘ·B͎̅₂ ,⋯, A⃗ₘ·B͎̅ₚ ⎦
= 
C⃣ₘₓₚ

Cᵢⱼ := A⃗ᵢ⋅B⃗ᵀⱼ


(A⃣B⃣)C⃣ = A⃣(B⃣C⃣)
(A⃣B⃣)𝑥⃗ = A⃣（B⃣𝑥⃗）

A⃣(B⃣+C⃣) = A⃣B⃣ + A⃣C⃣
(A⃣+B⃣)C⃣ = A⃣C⃣ + B⃣C⃣

Sqare Matrices

A⃣² = A⃣A⃣
A⃣ⁿ⁺ᵐ = A⃣ⁿ · A⃣ᵐ
(A⃣ⁿ)ᵐ = A⃣ⁿ⠁ᵐ

Invertion

A⃣⁻¹A⃣ = I⃣ = A⃣A⃣⁻¹
  det❨A⃣❩
  A⃣x⃗＝0⃗ ⟺ x⃗＝0⃗
(A⃣B⃣)⁻¹ = B⃣⁻¹ A⃣⁻¹

A⃣A⃣⁻¹＝I⃣

Transpose

(A⃣x⃗ )ᵀ = x⃗ᵀ A⃣ᵀ
(A⃣B⃣)ᵀ = B⃣ᵀ A⃣ᵀ
(A⃣⁻¹)ᵀ＝(A⃣ᵀ)⁻¹
A⃣𝑥⃗ ⋅ 𝑦⃗ = 𝑥⃗ ⋅ A⃣ᵀ𝑦⃗
(A⃣+B⃣)ᵀ =  A⃣ᵀ + B⃣ᵀ

Symmetric.

IFF S⃣ ＝S⃣ᵀ

Orthogonal

IFF Q⃣⁻¹＝Qᵀ

Permutation matrix

has the rows of I⃣ in any order.