commit 9c209bdc284af837c7d307f3094289d1b84def4c
parent 2b610dd03c0ac12b0dc25e269cad656d4c85101e
Author: Andrew <andrewlaack1@gmail.com>
Date: Wed, 10 Jul 2024 11:34:21 -0500
Took some notes
Diffstat:
16 files changed, 163 insertions(+), 1 deletion(-)
diff --git a/Calculus.md b/Calculus.md
@@ -5,3 +5,29 @@
## Main Links
[[NaturalLog.md]]
+[[ProductRule.md]]
+[[ChainRule.md]]
+[[LogarithmicDifferentiation.md]]
+
+
+## Known Integrals
+
+
+Trig Integrations:
+
+---
+
+sin(x) -> -cos(x) + c
+
+cos(x) -> sin(x) + c
+
+sec^2(x) -> tan(x) + c
+
+sec(x)tan(x) -> sec(x) + c
+
+csc(x)cot(x) -> -csc(x) + c
+
+csc^2(x) -> cot(x) + c
+
+---
+
diff --git a/ChainRule.md b/ChainRule.md
@@ -0,0 +1,8 @@
+:calc:
+# Chain Rule
+
+Leonard
+
+## Notes
+
+**Definition:** The chain rule is a derivation rule used when we have a function within another function. The rule states $\frac{d}{dx} (g(f(x))) = g'(f(x)) \cdot f'(x)$.
diff --git a/Complement.md b/Complement.md
@@ -0,0 +1,10 @@
+:prob:
+# Complement
+
+L1
+
+## Notes
+
+**Definition:** The complement of a set is the set of all elements not in the original set, but in the consideration space (often sample space).
+
+There are technically two types of complements the absolute and relative complements. Generally we are talking about the relative complement which is the set defined as the difference between the superset and the subset. The absolute complement uses the U set ([[UniversalSet.md]]) as the superset.
diff --git a/DiscreteUniformLaw.md b/DiscreteUniformLaw.md
@@ -0,0 +1,8 @@
+:prob:
+# Discrete Uniform Law
+
+L1
+
+## Notes
+
+**Definition:** The discrete uniform law states that if all outcomes in a [[SampleSpace.md]] are equally probable then P(A) where A is a set is the same as |A| / |Omega| where Omega is the entire sample space.
diff --git a/DisjointSet.md b/DisjointSet.md
@@ -0,0 +1,8 @@
+:prob:
+# Disjoint Set
+
+L1
+
+## Notes
+
+**Definition:** Disjoint sets are multiple sets where they have no elements in common.
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -33,3 +33,5 @@ The basis of linear algebra is solving systems of equations.
[[DistanceToPlane.md]]
[[PlaneToPlaneDistance.md]]
[[ReducedRowEchelonForm.md]]
+[[Transpose.md]]
+[[NullSpace.md]]
diff --git a/LogarithmicDifferentiation.md b/LogarithmicDifferentiation.md
@@ -0,0 +1,16 @@
+:calc:
+# Logarithmic Differentiation
+
+Leonard
+
+## Notes
+
+**Definition:** Logarithmic differentiation is the process of applying logs to both sides of an equation to aid in our ability to find their derivative.
+
+Steps:
+
+1. Take log of both sides
+2. Expand both sides (y will become d/dx(y) times 1/y)
+3. Solve derivative
+4. Multiply both sides by y to remove y term from left
+5. Replace y with original equation (or leave as is)
diff --git a/Matrix.md b/Matrix.md
@@ -25,3 +25,5 @@ The product of a matrix and a vector is another vector of the same size as the o
[a b c] [j] [aj bk cl]
[d e f] @ [k] = [dj ek fl] (note the result is a vector not a matrix)
[g h i] [l] [gj hk il]
+
+This can be thought of as the dot product of each row placed into the correct row as the only column.
diff --git a/NaturalLog.md b/NaturalLog.md
@@ -20,3 +20,11 @@ There are no other rules for manipulations so there are times when we are limite
d/dx ln(|x|) = 1/x
The |x| is only neccesary for functions that cross into the negative x values.
+
+### Chain Rule
+
+We can also use the chain rule for ln.
+
+Example:
+
+$\frac{d}{dx} (ln(g(x))) = \frac{1}{g(x)} \cdot g'(x)$
diff --git a/NullSpace.md b/NullSpace.md
@@ -0,0 +1,18 @@
+:lin-alg:
+# Null Space
+
+Khan
+
+## Notes
+
+**Definition:** The null space of matrix A is the set of vectors {$\vec{b} \in \R^n | \space \vec{b} \cdot A=\vec{0}$}.
+
+These are all of the vectors that when multiplied by the matrix are equivalent to the zero vector. This is a closed ([[Closure.md]]) [[Subspace.md]].
+
+To calculate the null space do the following:
+
+1. Get [[ReducedRowEchelonForm.md]]
+2. Write out find the values of each pivot entry (relation to other values)
+3. Plug this into vectors of height n where each vector is multiplied by the corresponding axis component
+
+The null set of a linearly independent set is always just the zero vector. This is an iff situation.
diff --git a/ProbabilityLaw.md b/ProbabilityLaw.md
@@ -0,0 +1,14 @@
+:prob:
+# Probabilty Law
+
+L1
+
+## Notes
+
+**Definition:** The probability law assigns some set A (event) a nonnegative P(A) that describes the likelihood fo the elements of A.
+
+The probability law specifies the likelihood of the input given the sample space. The rules for it are as follows:
+
+1. P(A) >= 0
+2. P(A union B) = P(A) + P(B) if A and B are [[DisjointSet.md]].
+3. P(Omega) = 1
diff --git a/ProductRule.md b/ProductRule.md
@@ -0,0 +1,8 @@
+:calc:
+# Product Rule
+
+Leonard
+
+## Notes
+
+**Definition:** The product rule is used when taking the derivative of two functions that are multiplied together. The rule is as follows $\frac{d}{dx}(g(x)f(x)) = g'(x)f(x) + f'(x)g(x)$
diff --git a/SampleSpace.md b/SampleSpace.md
@@ -0,0 +1,13 @@
+:prob:
+# Sample Space
+
+L1
+
+## Notes
+
+**Definition:** The sample space is the space of all possible outcomes of a random experiment.
+
+This should be two things
+
+1. Mutually Exclusive (No two results can happen during the same run)
+2. Collectively Exhaustive (No result can occur outside the domain of outcomes)
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -40,3 +40,14 @@ Links to Stats Notes
[[Percentile.md]]
[[CumulativeRelativeFrequency.md]]
[[IQR.md]]
+
+PSA&AP MIT:
+
+L1:
+ - [[SampleSpace.md]]
+ - [[Complement.md]]
+ - [[DiscreteUniformLaw.md]]
+ - [[UniversalSet.md]]
+ - [[DisjointSet.md]]
+ - [[ProbabilityLaw.md]]
+
diff --git a/Transpose.md b/Transpose.md
@@ -1,4 +1,4 @@
-:ml:
+:ml: :lin-alg:
# Transpose
ML P627
@@ -12,3 +12,5 @@ ML P627
4 6 3 1 2 3
As you can see, the first value remains and across the top we have the first column.
+
+Additionally, the transpose of a vector is possible and will go from n x 1 to 1 x n.
diff --git a/UniversalSet.md b/UniversalSet.md
@@ -0,0 +1,8 @@
+:prob:
+# Universal Set
+
+L1
+
+## Notes
+
+**Definition:** The universal set either denoted by U or Omega is the set of all objects that are of interest in a particular context.
