commit 499366d2d3d082dab76364d83b50c3adb3d419fb
parent 5f38dd08f86c7cc54c1a1e5de0517aeb4eb746bc
Author: Andrew <andrewlaack1@gmail.com>
Date: Thu, 11 Jul 2024 12:35:33 -0500
Took some notes
Diffstat:
8 files changed, 89 insertions(+), 7 deletions(-)
diff --git a/BasisOfSubspace.md b/BasisOfSubspace.md
@@ -12,3 +12,7 @@ Basically, we have a set of vectors and this is the basis of a subspace whereby
Note:
The basis is the **minimum** set of vectors to describe the span of the subspace. As such, all vectors must be linearly independent.
+
+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.
diff --git a/Bijective.md b/Bijective.md
@@ -0,0 +1,10 @@
+:calc:
+# Bijective
+
+L2
+
+## Notes
+
+**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).
diff --git a/Calculus.md b/Calculus.md
@@ -4,11 +4,19 @@
## Main Links
-[[NaturalLog.md]]
-[[ProductRule.md]]
-[[ChainRule.md]]
-[[LogarithmicDifferentiation.md]]
-
+Calc 2 (Leonard):
+
+L1:
+ - [[NaturalLog.md]]
+ - [[ProductRule.md]]
+ - [[ChainRule.md]]
+ - [[LogarithmicDifferentiation.md]]
+
+L2:
+ - [[InverseFunction.md]]
+ - [[Injective.md]]
+ - [[Surjective.md]]
+ - [[Bijective.md]]
## Known Integrals
diff --git a/Injective.md b/Injective.md
@@ -0,0 +1,8 @@
+:calc:
+# Injective
+
+L2
+
+## Notes
+
+**Definition:** For a function to be injective each value in the domain must map to a unique value in the codomain.
diff --git a/InverseFunction.md b/InverseFunction.md
@@ -0,0 +1,37 @@
+:calc:
+# Inverse Function
+
+L2
+
+## Notes
+
+**Definition:** The inverse function of f(x) is defined as f^-1(x) where f^-1(x) maps from the codomain of f(x) to the domain of f(x).
+
+As such, for a function to be invertible it must be a bijection.
+
+## Finding
+
+The simplest way to find the inverse of a function is by swapping the y and x terms and then solving for y.
+
+Consider:
+
+$$
+f(x) = x^3 + 10
+\newline
+y = x^3 + 10
+\newline
+
+$$
+
+$$
+x = y^3 + 10
+\newline
+x+10 = y^3
+\newline
+y = \sqrt[3]{x+10}
+$$
+
+$$
+f'(x) = \sqrt[3]{x+10}
+$$
+
diff --git a/ReducedRowEchelonForm.md b/ReducedRowEchelonForm.md
@@ -6,3 +6,6 @@ Khan
## Notes
**Definition:** Reduced row echelon form is a form of matrix where each row has a 1 after the zeoes that are all on the left side of the one. Additionally, each row above another row must have its 1 further to the left than the prior one, and all values to the right of the one should be zeroes if possible.
+
+
+If we are trying to find the basis of the column space then the columns with the pivot variables are that information. Alternatively, we can do RREF, write the equations, solve for the pivots based on the free variables. These will both give the correct result, but they will be different results as there are many statements of the basis of a span.
diff --git a/Surjective.md b/Surjective.md
@@ -0,0 +1,8 @@
+:calc:
+# Surjective
+
+L2
+
+## Notes
+
+**Definition:** For a function to be surjective each value in the codomain must be mapped to at least once.
diff --git a/TotalProbabilityTheroem.md b/TotalProbabilityTheroem.md
@@ -1,8 +1,12 @@
:prob:
-# Total Probability Theroem
+# Total Probability Theorem
L2
## Notes
-**Definition:** Total probability theroem states that the
+**Definition:** Total probability theorem states that the probability of some event is equal to the summed probability of each possible way for the event to occur.
+
+This is often done in the inverse where we have a bunch of conditions and want to find the probability of a given even occurring.
+
+**Reverses order of conditionals**
