commit 9f63a86e5e2238896646752c614fd8efd229d0ca
parent 4607a94f8672b92ba9ef4d870178896f8782b6b2
Author: Andrew <andrewlaack1@gmail.com>
Date: Wed, 7 Aug 2024 22:02:10 -0500
Took notes
Diffstat:
6 files changed, 126 insertions(+), 11 deletions(-)
diff --git a/:w b/:w
@@ -0,0 +1,47 @@
+:lin-alg:
+# Inverse Transformation (and matricies)
+
+Khan U2
+
+## Notes
+
+**Definition:** The inverse of a transformation is the transformation that undoes the original transformation for the entire domain codomain of the original transformation.
+
+This transformation must be [[Bijective.md]] otherwise there will be issues with mappings either there are outputs without inputs or there are outputs with multiple inputs.
+
+A transformation is invertible if and only if there exists an f^-1 such that f^-1 composed with f is I (identity function).
+
+(in the world of L.T.'s) Assume the standard matrix of f is A and the standard matrix of f^-1 is B. We then know:
+
+AB = I
+
+## Unique?
+
+Inverse functions are unique (there is only one).
+
+Let's assume they are not. We then find A(x) = B(x) for all x in R^n. This means A and B are the same function, but they are not. This is a contradiction.
+
+## Invertible?
+
+To find this we know it needs to be bijective. When solving for RREF if there are instances in the R^m space, where R^m is the codomain, that are not mapped to (found by having a row of zeroes where we can't map to everything based on the combination) then the standard matrix is not invertible as it stands from R^n to R^m.
+
+As such, T is onto iff C(A) = R^m (columns span R^m). **We know this is only true when RREF has a pivot in each row. ([[Rank.md]] of the matrix = m)**
+
+For injectivity we test for one-to-one. To find this we need to make sure the rank of the matrix is equal to n where n is the number of columns.
+
+Basically, we need a square matrix that is linearly independent (rows = columns).
+
+As such, the matrix is only invertable if the RREF is the identity matrix in R^n.
+
+## Solving
+
+To find the inverse of a matrix we find all of the manipulations done to the original matrix that results in us getting to the identity matrix. These manipulations can then be placed into their own matrix and thus derive the inverse.
+
+#### Example 1:
+
+$A = \begin{bmatrix}
+ 2 & 3 \\
+ 1 & 2
+\end{bmatrix}$
+
+
diff --git a/BernoulliProcess.md b/BernoulliProcess.md
@@ -5,6 +5,20 @@ Prob L13
## Notes
-**Definition:** A Bernoulli process is a sequence of binary trials.
+**Definition:** A Bernoulli process is a sequence of binary trials (random variables).
-Example is win/lose with a lottery ticket. Each trial is unrelated, but has the same probability (probably).
+As such, sample space are all possible sets of outcomes confined to a certain number of trials.
+
+Given that a bernoulli process must be memoryless, we can then derive that each trial must have equal probability otherwise each trial would not be independent.
+
+Ie.
+
+Sample space of 9 coin flips:
+
+000000000
+000000001
+.........
+.........
+.........
+111111110
+111111111
diff --git a/InverseTransformation.md b/InverseTransformation.md
@@ -35,4 +35,36 @@ As such, the matrix is only invertable if the RREF is the identity matrix in R^n
## Solving
+#### Intuitive
+
+There are formulas and stuff for solving this, but let's think from an intuitive sense first. To solve for the inverse of some matrix we need to find the matrix that converts from the original matrix to the identity matrix. If we are to create an augmented matrix where the left is the original matrix and the right is the identity matrix we can then perform changes to both sides to convert the left side to the identity matrix. By doing this we will then be left with the right side as the inverse matrix.
+
+That's a lot of text so let's do a simple example of a 2x2 matrix:
+
+$A=\begin{bmatrix}
+2 & 3 &| 1 & 0 \\
+1 & 2 & |0 & 1
+\end{bmatrix}$
+
+$\begin{bmatrix}
+1 & 1 & |1 & -1 \\
+0 & 1 &|-1 & 2
+\end{bmatrix}$
+
+$\begin{bmatrix}
+1 & 0 & |2 & -3 \\
+0 & 1 & |-1 & 2
+\end{bmatrix}$
+
+Solution:
+
+$A^{-1} =\begin{bmatrix}
+2 & -3 \\
+-1 & 2
+\end{bmatrix}$
+
+As can be seen we simply make changes on the left side and update the former identity accordingly to track what matrix would result in such a matrix on the left side starting from the original matrix.
+
+#### Formulaic
+
diff --git a/MarkovChains.md b/MarkovChains.md
@@ -0,0 +1,12 @@
+:prob:
+# Markov Chains
+
+L13
+
+## Notes
+
+**Definition:** A markov chain is a sequence of events where the probability of any given event is entirely based on the previous event.
+
+Anything that evolves with time can be described as a markov chain.
+
+These types of processes are not memoryless like [[BernoulliProcess.md]] or [[PoissonDistribution.md]].
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -90,3 +90,4 @@ L12:
L13:
- [[BernoulliProcess.md]]
- [[PoissonDistribution.md]]
+ - [[MarkovChains.md]]
diff --git a/index.md b/index.md
@@ -3,7 +3,9 @@
This is the index for my main note classifications. I will maintain this as a home page.
-## School
+[[ModelNotes.md]]
+
+## Formal Schooling
[[CS202.md]]
[[CS331.md]]
@@ -12,22 +14,29 @@ This is the index for my main note classifications. I will maintain this as a ho
[[Math310.md]]
[[TexRef.md]]
-## Other
+## Other Focuses
[[ComputerArchitecture.md]]
[[MachineLearning.md]]
[[StatisticsAndProbability.md]]
[[LinuxStuff.md]]
-[[ModelNotes.md]]
[[LinearAlgebra.md]]
[[Calculus.md]]
[[Algorithms.md]]
-## Technology Books to Read
+## Things to Learn More About
+
+Stats + Prob
+ - [ ] Markov Chains
+ - [ ] ECDF (sort of like cdf)
+ - [ ] Convolution (not NN)
-- [ ] "Introduction to Statistical Thought" - Michael Lavine
-- [ ] "Hands-On Machine Learning with Scikit-Learn and TensorFlow" - Aurelien Geron
-- [ ] "Introduction to Linear Algebra" - Gilber Strang
-- [ ] "Calculus Early Transcendentals" - James Stewart
+Lin-alg
+ - [ ] Inverse Matrix Calculations (do some practice problems)
-Maybe book about NN, transformers, cv, pytorch, I don't know what I will be missing by the end of the current ml book I am reading.
+ML
+ - [ ] RNN
+ - [ ] LSTM
+ - [ ] Mamba
+ - [ ] Transformer
+ - [ ] KAN
