commit 72507e3f28d0dbfac3e0d19e5537de5a6a9ba817
parent 61466139b20898c6abd75c38a599727eb420645f
Author: Andrew <andrewlaack1@gmail.com>
Date: Sat, 24 Aug 2024 12:18:01 -0500
Took some notes on a bunch of stuff
Diffstat:
11 files changed, 129 insertions(+), 6 deletions(-)
diff --git a/Algorithms.md b/Algorithms.md
@@ -27,6 +27,9 @@ L4:
L5 (non-comparative sorting):
- [[CountSort.md]]
+L6:
+ - [[BinaryTree.md]]
+
#### Intro To Algorithms Textbook (CLRS)
2.1
@@ -42,6 +45,3 @@ L5 (non-comparative sorting):
- [[AsymptoticNotation.md]]
- [[Trichotomy.md]]
- [[MonotonicFunction.md]]
-
-5.2
- - [[Relation.md]] - TODO
diff --git a/Bijective.md b/Bijective.md
@@ -8,3 +8,5 @@ L2
**Definition:** For a function to be bijective it must be both [[Surjective.md]] and [[Injective.md]].
This means that each value in the domain maps to a unique value in the codomain (Injective) and each value in the codomain is mapped to at least once (Surjective).
+
+Note: Another term for a bijection is one-to-one correspondence because injection is sometimes called one-to-one and a surjection is sometimes called onto.
diff --git a/BinaryTree.md b/BinaryTree.md
@@ -1,4 +1,4 @@
-:cs202:
+:cs202: :data-structures:
# Binary Tree
CS202 L14
@@ -9,4 +9,12 @@ CS202 L14
For a generic binary tree, there is no necessitation that the left a right trees are in any way balanced.
-A balanced binary tree has a logn time complexity
+A balanced binary tree has search time complexity of logn.
+
+## Datastructure Specifics
+
+Depth - Number of edges from the current node to the root of the tree
+
+Height - Number of edges for the longest downward path
+
+Subtree - A tree that is defined as subtree(a) where each node below a is included in the subtree
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -72,6 +72,7 @@ Unit 2.3 (functions):
- [[InverseFunction.md]]
- [[Floor.md]]
- [[Ceiling.md]]
+ - [[Bijective.md]]
Unit 2.4 (sequence + other stuff):
- [[Sequence.md]]
diff --git a/EigenVector.md b/EigenVector.md
@@ -10,3 +10,73 @@ Self Study
Associated with this, we also have an Eigen value which is the amount that a point on the Eigen Vector is distorted by (multiplied by this scalar)
This can be thought of as the axis of rotation when in R^3. Additionally, we know the eigen value is 1 in a rotation as there is no stretching.
+
+Formula:
+
+Where T(x) is a L.T., A is L.T.'s matrix, v is a vector, lambda is a scalar
+
+T(v) = Av = lambda v
+
+There are eigen vectors iff det(lambda I_n - A) = 0
+
+## Calculation
+
+A = [1 2]
+ [4 3]
+
+Eigen Value Calculation:
+
+det(lambda [1 0] - [1 2]) = 0
+ [0 1] [4 3]
+
+det([lambda 0] - [1 2]) = 0
+ [0 lambda] [4 3]
+
+det([lamda-1 -2]) = 0
+ [-4 lambda-3]
+
+(lambda-1) (lambda-3) - 8 = 0
+
+lambda^2 - 4lambda - 5 = 0
+
+(lambda-5)(lambda + 1) = 0
+
+Solutions:
+
+lambda = 5 or lambda = -1
+
+(lambda = eigen value)
+
+Eigen Vector Calculation (calculating for 5):
+
+0 = (lambda I_n - A) v
+
+= ([5 0] - [1 2] ) v
+ [0 5] [4 3]
+
+= [4 -2] v
+ [-4 2]
+
+Null space calculation:
+
+[4 -2]
+[-4 2]
+
+[4 -2]
+[0 0]
+
+[1 -1/2][v_1] = [0]
+[0 0][v_2] [0]
+
+v_1 - 1/2v_2 = 0
+
+v_1 = 1/2 v_2
+
+E_5 + {[v_1] = t[1/2], t \in \R}
+ [v_2] [1 ]
+
+Answer:
+span([1/2])
+ [ 1]
+
+The other calculation would be the same but for -1 thus it will be left as an exercise for the reader.
diff --git a/GramSchmidtProcess.md b/GramSchmidtProcess.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# The Gram-Schmidt Process
+
+Khan U3
+
+## Notes
+
+**Definition:** The Gram-Schmidt process is a process for finding an orthonormal basis of a subspace.
+
+Basically, if we have a basis we can find the orthonormal basis of the subspace by normalizing the first vector, projecting a subsequent one onto it and subtracting that from the original vector, normalizing that new vector, and repeating each time projecting onto all existing basis'.
diff --git a/Image.md b/Image.md
@@ -20,3 +20,9 @@ Ex.
T(V) = image of V under T
We call this the image of T stated as im(T) when we are referring to any vector in R^n not necessarily a given subspace. The distinction here is that T(V) defines V as the codomain whereas im(T) defines the codomain as all possible vectors in R^n.
+
+## Set Notation
+
+The image of the set S under f is defined as follows:
+
+$img(S) = f(S) = \{f(s) | s \in S\}$
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -78,3 +78,5 @@ Khan Unit 3:
- [[Projection.md]]
- [[ChangeOfBasis.md]]
- [[Orthonormal.md]]
+ - [[GramSchmidtProcess.md]]
+ - [[EigenVector.md]]
diff --git a/StatisticalInference.md b/StatisticalInference.md
@@ -0,0 +1,8 @@
+:: :: :: :: ::
+#
+
+
+
+## Notes
+
+**Definition:**
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -9,6 +9,22 @@ Links to Stats Notes
## Main Links
+Probability and Statistical Inference Hogg, Tanis:
+
+Chapter 1.1:
+ - [[SampleSpace.md]]
+ - [[StatisticalInference.md]] - TODO
+ - [[Frequency.md]] - TODO
+ - [[RelativeFrequency.md]] - TODO
+ - [[ProbabilityMassFunction.md]]
+ - [[SimpsonsParadox.md]] - TODO
+ - [[RandomExperiment.md]] - TODO VS Random Variable
+
+Chapter 1.2:
+ - [[Event.md]] - TODO
+
+---
+
[[Probability.md]]
[[SetFunction.md]]
[[MonotonicFunction.md]]
diff --git a/index.md b/index.md
@@ -48,4 +48,4 @@ ML
- [ ] Linear Regression statistical approach
Algorithms
- - [ ]
+ - [ ] Fisher-Yates (Knuth) shuffle algorithm
