commit 83aa7f7f7b76f1a7a62376a0078b7e636efd1cdd
parent f3fe72750c97377149734db321b0f166ada60793
Author: Andrew <andrewlaack1@gmail.com>
Date: Fri, 9 Aug 2024 17:19:06 -0500
Took some notes
Diffstat:
7 files changed, 116 insertions(+), 7 deletions(-)
diff --git a/Determinant.md b/Determinant.md
@@ -1,11 +1,43 @@
-:cs331: :math: :linear-algebra:
-# Determinanat
+:cs331: :math: :lin-alg:
+# Determinant
-CS331 - Linear Algebra
+CS331 - Linear Algebra - Khan U2
## Notes
**Definition:** The determinant is the scaling factor of some area (or volume in 3d space) from before to after a linear transformation.
-
This value can be negative if the space has been flipped. In 3d space, this means the volume after the tranformation is in left hand space if it was before in right hand space.
+
+Note: If the determinant of the matrix is 0 then the matrix is not invertible.
+
+## Calculation
+
+#### 2x2
+
+For a 2x2 matrix we have Det(A) = ad-bc where the matrix A is defined as <a,b> <c,d> where each vector is a row.
+
+Ex:
+
+A = [1 2]
+ [3 4]
+
+Det(A) = 4 - 6 = -2
+
+#### Generalized
+
+For each element in the first row of the matrix remove all values in the same row and column and then multiply the current element by the determinant of the remaining elements.
+
+We then add all of the determinant element pairs where every other is negative (first is positive second negative and so on).
+
+This is called Laplace expansion or cofactor expansion.
+
+Note: You don't need to use the first row as the row you traverse to find the determinant. The only thing that matters is that the first row starts with a + and switches back and forth thus if you pick the last row to evaluate with you might start with a negative depending on the size of the matrix. It often makes sense to do this because it allows for selection of a row with many zeroes.
+
+With this, you are not stick layer to layer with a given row either so you can select the last row for the largest matrix and the second for a smaller and so on.
+
+#### Scalar
+
+When multiplying a **row** by a scalar the resulting determinant is equal to the original determinant multiplied by the scalar.
+
+This can be extrapolated to the size of a matrix (matrix times scalar) and thus it becomes dependent upon the number of rows in the matrix.
diff --git a/InverseTransformation.md b/InverseTransformation.md
@@ -33,6 +33,8 @@ 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.
+Another thing about this; if the determinant is not 0 the matrix is invertible if not then it is not invertible.
+
## Solving
#### Intuitive
@@ -67,4 +69,22 @@ As can be seen we simply make changes on the left side and update the former ide
#### Formulaic
+The formula is useful, but simply a description of the rules we can apply. For the 2x2 formula we 0 the first column from the second row, zero the second column from the first row, and then normalize both rows.
+
+I was going to show this derivation, but it is trivial and long.
+
+##### 2x2 Formula:
+
+$A =\begin{bmatrix}
+a & b \\
+c & d
+\end{bmatrix}$
+
+$A^{-1}=
+\frac{1}{ad-bc}
+\begin{bmatrix}
+d & -b \\
+-c & a
+\end{bmatrix}$
+The interestinge thing about this is the denominator of the fraction out front is the determinant of the matrix.
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -61,3 +61,5 @@ Khan Unit 2:
- [[Bijective.md]]
- [[Homogeneous.md]]
- [[Inhomogeneous.md]]
+ - [[Determinant.md]]
+ - [[RuleOfSarrus.md]]
diff --git a/PoissonProcess.md b/PoissonProcess.md
@@ -0,0 +1,16 @@
+:prob:
+# Poisson Process
+
+Prob L14
+
+## Notes
+
+**Definition:** A poisson process is a continous time version of the [[BernoulliProcess.md]].
+
+A poisson process models continuous time with binary outcomes. Generally, we simply track when the true case occurs.
+
+Poisson processes presuppose independence and homogenity of probability over time.
+
+## See Also
+
+[[BernoulliProcess.md]] - Memoryless, discrete time, process of binary outcomes
diff --git a/RuleOfSarrus.md b/RuleOfSarrus.md
@@ -0,0 +1,31 @@
+:lin-alg:
+# Rule of Sarrus
+
+Khan U2
+
+## Notes
+
+**Definition:** The rule of Sarrus is a shortcut for finding the determinant of a 3x3 matrix.
+
+Formula:
+
+ [a b c]
+Det ([d e f]) = aei + bfg + cdh - afh - bhi - ceg
+ [g h i]
+
+When looking at the matrix we add the multiplied diagonals (starting from top row 3 values) to the right and subtract the diagonals to the left (multiplying each value in the diagonal).
+
+See [[Determinant.md]] for calculating determinants, what they represent, and how to find 2x2 with a formula.
+
+Ex:
+
+ [1 2 4]
+A = [2 1 3]
+ [3 4 8]
+
+
+Det(A) = 1x1x8 + 2x3x3 + 4x2x4 - 1x3x4 - 2x2x8 - 4x1x3
+
+= 8 + 18 + 32 - 12 - 32 - 12
+
+= 2
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -88,6 +88,9 @@ L11:
L12:
- [[IteratedExpectations.md]]
L13:
+ - [[BernoulliProcess.md]] - Discrete memoryless
+ - [[MarkovChains.md]] - Discrete remembers
+ - [[PoissonProcess.md]] - Continuous memoryless
+L14:
+ - [[PoissonProcess.md]]
- [[BernoulliProcess.md]]
- - [[PoissonDistribution.md]]
- - [[MarkovChains.md]]
diff --git a/index.md b/index.md
@@ -30,10 +30,15 @@ Stats + Prob
- [ ] Markov Chains
- [ ] ECDF (sort of like cdf)
- [ ] Convolution (not NN)
+ - [ ] https://en.wikipedia.org/wiki/Primality_test
+ - [ ] https://en.wikipedia.org/wiki/Continuous-time_Markov_chain
Lin-alg
- [ ] Inverse Matrix Calculations (do some practice problems)
-
+ - [ ] https://en.wikipedia.org/wiki/Invertible_matrix
+ - [ ] Laplace Expansion (define outside of determinant note)
+ - [ ] Cofactor (define)
+ - [ ] Minors
ML
- [ ] RNN
- [ ] LSTM
