Probability, Linear Algebra and Differential Equations

• This text appears to be study notes for a mathematics or related course, covering topics from classical and geometric probability models to differential equations.
• It also includes linear algebra basics like matrices, vector spaces, and linear transformations.

Classical Probability Model

• Basic events are mutually exclusive.
• Any event (except an impossible event) is the sum of basic events.
• A classical probability model having these two characteristics:
  • Limited number of all possible outcomes of the experiment, and only one happens at a time
  • Each result has an equal chance of happening.
• In an experiment with n possible equally likely outcomes, the probability of each basic event is 1/n.
• For an event A with m outcomes, the probability of A is P(A) = m/ n.

Geometric Probability Model

• A geometric probability model is one where the probability of an event is proportional to the length, area, or volume that constitutes the event’s region.
• Key properties of a geometric probability model:
• Unlimited: There could be endless results from the experiment it models
• Equally likely: all results have the same probability
• If there are multiple, use P(A) = (Measure of region constituting event A)/(Measure of region constituting all possible outcomes).

L'Hôpital's Rule (洛必达法则)

• When both the numerator and denominator have derivatives, use derivation until a determinate form is solvable by direct substitution.
• Used in indeterminate forms like 0/0 or ∞/∞.

Methods for ∞/∞ or 0/0 limits

• Simplify the expression by canceling factors that approach zero or infinity.
• L'Hôpital's Rule (洛必达法则) can be used.
• Substitute variables and use important limits.
• Use infinitesimal equivalents.

Limits with the form 0*∞

• Convert the expression to 0/0 or ∞/∞ to apply L'Hôpital's Rule.
• Generally, factors that are more complex should be put into the numerator
• Usually factors with logarithms are placed in the numerator.

Limits with the form ∞-∞

• Convert them by appropriate form to 0/0 or ∞/∞ form
• Convert by finding common denominators or rationalizing numerators

Important Limits

• lim (x->0) sinx/x = 1
• lim (x->∞) (1 + 1/x)^x = e
• lim (x->0) (1 + x)^(1/x) = e

Convergence and Divergence of Series

• A series Σuᵢ converges if the sequence of its partial sums {Sₙ} has a limit S
• The value S is the sum of a convergent series.
• If it meets this criteria, then it will diverge
• The term rₙ = S - Sₙ is the "remainder" or "residual sum" of this series.

Geometric Series

• Has the form Σqⁿ
• Converges if |q| < 1.
• Diverges if |q| ≥ 1

p-Series

• Has the form Σ 1/n^p
• Converges if p > 1.
• Diverges if p ≤ 1.

Properties of Numerical Series

• If Σuₙ converges to S, then Σkuₙ converges to kS, where k is a constant.
• If Σuₙ converges to α and Σvₙ converges to β, then Σ(uₙ ± vₙ) converges to α ± β.
• Adding, removing, or changing a finite number of terms does not affect convergence or divergence.

Squeeze Theorem (两边夹定理)

• For series: If uₙ ≤ vₙ ≤ wₙ and Σuₙ and Σwₙ both converge to a, then Σvₙ also converges to a.

Convergence Condition

• If Σuₙ converges, then lim (n→∞) vₙ = 0

Cauchy Convergence Criterion (柯西收敛原理)

• Σuₙ converges if and only if for every ε > 0, there exists N such that for n > N and any integer p, |uₙ₊₁ + uₙ₊₂ + ... + uₙ₊ₚ| < ε.

Tests for Convergence of Positive Term Series

• If uₙ ≥ 0 (n=1,2,...) which makes Σuₙ a postive term
• Sufficient and Necessary condition, with the partial amount being its array is what makes it converge

Comparison Test

• If 0 ≤ uₙ ≤ vₙ, then the series will either diverge or converge together

Ratio Test (比值法)

• Determine if ∑uₙ converges with:
• ρ = lim (n→∞) uₙ₊₁/uₙ
• if ρ < 1 then the series will converge
• if ρ > 1 or lim (n→∞) uₙ₊₁/uₙ = ∞, then the series will diverge.
• If ρ = 1 then you cannot determine the divergence of convergence

Root Test (根值法)

• If lim (n→∞) ⁿ√uₙ = ρ, then:
  • If ρ < 1, ∑uₙ converges
  • If ρ > 1, ∑uₙ diverges
  • If ρ = 1, the test is inconclusive

Alternating Series Convergence

• Leibniz's Theorem: An alternating series Σ(-1)ⁿ⁻¹uₙ converges if:

  • uₙ ≥ uₙ₊₁ (n = 1, 2, 3, ...)
  • lim (n→∞) uₙ = 0

• If this series is converging, then its divergence is less than the first, s ≤ u1, absolute value is |rₙ| ≤ uₙ+1

• Absolute and Conditional Convergence: It is required that |uₙ| is convergent to the same

Properties of Definite Integrals

1. ∫ₐᵃ f(x) dx = 0
2. ∫ₐᵇ 1 dx = b - a
3. ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx
4. ∫ₐᵇ kf(x) dx = k ∫ₐᵇ f(x) dx
5. ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫cᵇ f(x) dx
6. ∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx
7. If f(x) ≥ 0 on [a, b], then ∫ₐᵇ f(x) dx ≥ 0
8. If f(x) ≤ g(x), then ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx
9. |∫ₐᵇ f(x) dx| ≤ ∫ₐᵇ |f(x)| dx
10. If m ≤ f(x) ≤ M on [a, b], then m(b-a) ≤ ∫ₐᵇ f(x) dx ≤ M(b-a)
11. Mean Value Theorem for Integrals: There exists ξ in [a, b] such that ∫ₐᵇ f(x) dx = f(ξ)(b-a)
12. If f(x) is odd, then ∫₋ₐᵃ f(x) dx = 0 ; If f(x) is even, then ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx

Integrals with Variable Limits

• F(x) = ∫ₐ₍ₓ₎ᵇ₍ₓ₎ f(t) dt
• F'(x) = f(b(x))b'(x) – f(a(x))a'(x)

Basic Properties of Determinants

• The value of a determinant equals its transpose.
• Swapping two rows or columns changes the sign of the determinant.
• If two rows or columns are identical, the determinant is zero.
• If a row or column has a common factor k, it can be factored out.
• If a row or column consists entirely of zeros, the determinant is zero.
• If two rows or columns are proportional, the determinant is zero.

Matrix Concepts

• A matrix A of size m×n over a field F is an array of mn elements.
• Equal Matrices: A = B if corresponding elements are equal, where aᵢⱼ = bᵢⱼ(i = 1, 2, ..., s; j = 1, 2, ..., n)
• An n-order square matrix is an A = (aᵢⱼ)m×n matrix when m = n.
• Zero Matrix: A matrix where all elements are 0.
• Symmetric Matrix: A matrix A, if aᵢⱼ = aⱼᵢ.
• Skew-Symmetric Matrix: A matrix A, if then aᵢⱼ = -aⱼᵢ
• Diagonal elements of a skew-symmetric matrix are zero.

Triangular Matrices

• An upper triangular matrix is when all elements are zero in the lower corner
• A lower triangular matrix is when all elements are zero in the upper corner

Diagonal Matrix

All the square elements are zero except the diagonal matrices elements

Scalar Matrix

Where A = diag(a₁₁, a₂₂, ..., aₘₙ) describes elements scaling with the appropriate scaling factor

Matrix Addition

• Sum the component matrices together to compute C = (aᵢⱼ + bᵢⱼ)
• Follows the same functions of addition, commutative properties as the original
• Commutative: A + B = B + A
• Associative: (A + B) + C = A + (B + C)
• A + 0 = 0 + A = A
• A + (-A) = 0

Scalar Multiplication

• When k is described as a scalar
• Where the matrix A can be defined as (aᵢⱼ) multiplied by matrix A's scalar amount, such as k(aᵢⱼ) where k is not already in Matrix A

Matrix Multiplication

• With two matrix inputs, such as A = (aᵢₖ)sxm, B = (bₖⱼ)mxn
• Then we can multiply matrix A with matrix B where C with the subscript (cᵢⱼ)sxn
• Σ aₖb[1];i + a[1];2b[2];i +... + a[I][end]b[end];i, or equal to a[I][K]b[K][J], which is the new Matrix C

Operations

  - Associative (AB)C = A(BC.
  - Distributive (A+B)C = AC+BC /  C(A+B) = CA + CB
  - k(AB) = (kA)B = A(kB)
  - kA = (KE)A = A_(KE)

Eigen Values in Space

  - Aξ = λξ.  Where A is an Eigen Value when a non zero vector to is defined, which is a set of Eigen Vectors on A.
  -  λE - A called Eigen Matrix

Linear Space

  - Non Vector space V w/ scaling values of vectors , with value k , defined as Linear Space. This meets:  (1α + β= β + α. ②(α + β + γ = a + (β + γ).

③0 + α = α. ④α − α = 0. ⑤1α = α. ⑥k(la) =(kl)a.

Vector Linear space. Has to be equal to :

①零元素是唯一的 (all zeroes) ②负元素是唯一的 ③0α = 0; k0 = 0; (−1)α = −α ④如果ka = 0,那么k = 0或α = 0

High Linearly independent

① 线性组合. If one can be written as vector quantity , aka Linear Combination ②线性相(无)关. If one can be written as vector quantity , aka Linearly Dependent

High Linearly independent

① 线性组合. This vector quantity allows us to combine the linear aspects. So we can linearly depict.

Two Planes Relation Position:

If

• If ₁ ∥ ₂, and all of the constants are linearly respective.
• If ₁ ⊥ ₂, cross sections dont equate to zero

Two Vector Relations:

• L ₁= L₂
• All must equate, including x y z constraints.

Plane Vector value Calculation:

Must not be above 90 angle to be valid cos value to the direction cos θ = √²+m²+n² √√1+1=

Differentail Vector Equations

Can be shown as y' = f(x)g(y) ∴ differential Equations is what is used to seprate Then integrate at g(y) dy Then intgrate dx to get the equation