commit 09f92a3d4660568ec411624e9a668eae5844ddd9
parent 499366d2d3d082dab76364d83b50c3adb3d419fb
Author: Andrew <andrewlaack1@gmail.com>
Date: Sat, 13 Jul 2024 19:40:16 -0500
took some notes
Diffstat:
8 files changed, 79 insertions(+), 0 deletions(-)
diff --git a/BasisOfSubspace.md b/BasisOfSubspace.md
@@ -16,3 +16,7 @@ The basis is the **minimum** set of vectors to describe the span of the subspace
This can be found two ways. The first is to find RREF of the matrix, write it out as equations, solve for the pivot variables values in relation to the free variables, write this out in vector coeficcient multiplication form, and then the columns we have are our answers.
Alternatively, we can find the pivot variables using RREF and these columns will contain the basis of the subspace. This is similar to what was done above except is uses the original columns instead of derived columns to state the basis.
+
+### Intuitive
+
+The intuitive idea is this is a linearly independent set of vectors. As such, there are infinite basises for a given subspace, but they all have the same cardinality.
diff --git a/BinomialCoefficient.md b/BinomialCoefficient.md
@@ -0,0 +1,28 @@
+:prob:
+# Binomial Coefficient
+
+L4
+
+## Notes
+
+**Definition:** A binomial coefficient is represented by two numbers and has a singular evaluation. The evaluation describes the number of unique subsets of the length denoted by the bottom value that can be created given a set of the length denoted by the top value.
+
+### Formula
+
+(n) = n! / ((r!(n-r)!)
+(r)
+
+### Example
+
+(8) = 8! / ((3!(8-3)!)) = 40320 / (6 x 120) = 40320/720 = 56
+(3)
+
+8 choose 3 is 56
+
+### Intuition
+
+The top of the function is all permutations of the list. The problem with this is that it includes rearrangements which we don't care for and because it is limited to the length of the entire set.
+
+As such, we divide this by r! to account for the arrangements of the r items. The second part, (n-r!) accounts for arrangements where we are not choosing r items.
+
+All together, we find the permutations of sets length n then divide this by r! to find the number of distinct sets not arrangements and then we divide by (n-r)! to get rid of sets that don't have r items.
diff --git a/Independence.md b/Independence.md
@@ -0,0 +1,19 @@
+:prob:
+# Independence
+
+L3
+
+## Notes
+
+**Definition:** Independence in probability is the case where some even B occuring does not affect the conditional probability of A occuring.
+
+Two Formal Definitions:
+
+1. P(A|B) = P(A)
+2. P(A intersect B) = P(A)P(B)
+
+Second definition is better, but first, to me, is more intuitive.
+
+## Conditional Independence
+
+There can also be independence given conditions called conditional independence. This occurs when specifying a condition and finding out that our two conditional definitions are true in the newly refined sample space that has been updated with further knowledge.
diff --git a/LatentSpace.md b/LatentSpace.md
@@ -0,0 +1,10 @@
+:ml:
+# Latent Space (embedding space)
+
+SS
+
+## Notes
+
+**Definition:** Latent space is a lower dimensional embedding space used to represent higher dimensional information.
+
+Think of using an autoencoder to embed high dim data into lower dimensions. This ties in more generally with dimensionallity reduction as well.
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -35,3 +35,4 @@ The basis of linear algebra is solving systems of equations.
[[ReducedRowEchelonForm.md]]
[[Transpose.md]]
[[NullSpace.md]]
+[[Nullity.md]]
diff --git a/MachineLearning.md b/MachineLearning.md
@@ -161,6 +161,9 @@ Concepts:
[[PoolingLayers.md]]
[[DataAugmentation.md]]
[[SMOTE.md]]
+[[LatentSpace.md]]
TODO:
+[[T-SNE.md]]
+[[UMAP.md]]
[[MCTS.md]]
diff --git a/Nullity.md b/Nullity.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Nullity
+
+Khan
+
+## Notes
+
+**Definition:** The nullity of a matrix is the dimensionallity of its [[NullSpace.md]].
+
+The nullity of a matrix is equal to the number of non-pivot (free) variable columns.
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -55,3 +55,7 @@ L2:
- [[BayesTheroem.md]]
- [[TotalProbabilityTheroem.md]]
- [[ConditionalProbabilityTheroem.md]]
+L3:
+ - [[Independence.md]]
+L4:
+ - [[BinomialCoefficient.md]]
