commit 4607a94f8672b92ba9ef4d870178896f8782b6b2
parent 364e2a5f6eddf7fb85af5942034d4c33f69bfd05
Author: Andrew <andrewlaack1@gmail.com>
Date: Tue, 6 Aug 2024 19:10:35 -0500
took linalg + stats notes
Diffstat:
8 files changed, 56 insertions(+), 1 deletion(-)
diff --git a/BernoulliProcess.md b/BernoulliProcess.md
@@ -0,0 +1,10 @@
+:prob:
+# Bernoulli Process
+
+Prob L13
+
+## Notes
+
+**Definition:** A Bernoulli process is a sequence of binary trials.
+
+Example is win/lose with a lottery ticket. Each trial is unrelated, but has the same probability (probably).
diff --git a/Homogeneous.md b/Homogeneous.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Homogeneous
+
+Khan U2
+
+## Notes
+
+**Definition:** In linear algebra a homogeneous solution is one where the right side of the system is the zero vector.
+
+See also [[Inhomogeneous.md]]
diff --git a/Inhomogeneous.md b/Inhomogeneous.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Inhomogeneous
+
+Khan U2
+
+## Notes
+
+**Definition:** An inhomogeneous solution in linear algebra is a solution where the right side of the system of equations is not the zero vector.
+
+See also [[Homogeneous.md]]
diff --git a/Injective.md b/Injective.md
@@ -6,3 +6,7 @@ L2
## Notes
**Definition:** For a function to be injective each value in the domain must map to a unique value in the codomain.
+
+Sometimes called one-to-one because there is one y for each x.
+
+For any y in Y there is at most one x such that f(x) = y.
diff --git a/InverseTransformation.md b/InverseTransformation.md
@@ -21,4 +21,18 @@ 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.
-## Calculation
+## 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
+
+
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -59,3 +59,5 @@ Khan Unit 2:
- [[Surjective.md]]
- [[Injective.md]]
- [[Bijective.md]]
+ - [[Homogeneous.md]]
+ - [[Inhomogeneous.md]]
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -87,3 +87,6 @@ L11:
- [[CorrelationCoefficient.md]]
L12:
- [[IteratedExpectations.md]]
+L13:
+ - [[BernoulliProcess.md]]
+ - [[PoissonDistribution.md]]
diff --git a/Surjective.md b/Surjective.md
@@ -6,3 +6,5 @@ L2
## Notes
**Definition:** For a function to be surjective each value in the codomain must be mapped to at least once.
+
+Also known as **"onto"**
