commit e570d7d69e66895c804db5aeee31208636c3031a
parent 7310b0d1029adee2d45f135cf222888e54b82d0c
Author: Andrew <andrewlaack1@gmail.com>
Date: Wed, 21 Aug 2024 20:19:14 -0500
Took discrete notes
Diffstat:
8 files changed, 103 insertions(+), 1 deletion(-)
diff --git a/Ceiling.md b/Ceiling.md
@@ -0,0 +1,12 @@
+:discrete:
+# Ceiling
+
+U2.3.4
+
+## Notes
+
+**Definition:** The ceiling function specifies to round up the input to the nearest integer.
+
+Remember to still round to the higher number for negatives.
+
+$\lceil 10.1 \rceil = 11$
diff --git a/ChangeOfBasis.md b/ChangeOfBasis.md
@@ -0,0 +1,30 @@
+:lin-alg:
+# Change of Basis
+
+Khan U3
+
+## Notes
+
+**Definition:** Change of basis in linear algebra is the process of assuming the basis vectors to be some arbitrary linearly independent vectors.
+
+Example:
+
+B = { [1] [2]
+ [2] [1]}
+
+a = 3B_1 + 2B_2
+
+[a]\_B = [3]
+ [2]
+
+While we have stated a to be [3 2] we are assigning it with basis' of B so in the standard coordinate system a = [8 7].
+
+## Matrix Representation
+
+The matrix representation of a change of basis is simply a matrix that we multiply all matricies under the basis by to find the true coordinates using the new basis'.
+
+The matrix representation of a change of basis is always invertible.
+
+## L.T.s
+
+Linear transformations are specified under the basis they are being applied to and do not apply under different basis'.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -68,3 +68,10 @@ Unit 2.3 (functions):
- [[Codomain.md]]
- [[Injective.md]] - one-to-one
- [[Surjective.md]] - onto
+ - [[InverseFunction.md]]
+ - [[Floor.md]]
+ - [[Ceiling.md]]
+
+Unit 2.4 (sequence + other stuff):
+ - [[Sequence.md]]
+ - [[RecurrenceRelation.md]]
diff --git a/Floor.md b/Floor.md
@@ -0,0 +1,12 @@
+:discrete:
+# Floor
+
+U2.3.4
+
+## Notes
+
+**Definition:** The floor function specifies to round down the input to the nearest integer.
+
+Remember to round to the lower number for negative numbers.
+
+$\lfloor 10.9 \rfloor = 10$
diff --git a/InverseFunction.md b/InverseFunction.md
@@ -1,4 +1,4 @@
-:calc:
+:calc: :discrete:
# Inverse Function
L2
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -76,3 +76,4 @@ Khan Unit 2:
Khan Unit 3:
- [[OrthogonalComplement.md]]
- [[Projection.md]]
+ - [[ChangeOfBasis.md]]
diff --git a/RecurrenceRelation.md b/RecurrenceRelation.md
@@ -0,0 +1,14 @@
+:discrete:
+# Recurrence Relation
+
+U2.4.2
+
+## Notes
+
+**Definition:** A recurrence relation is an equation that expresses some a_n in terms of one or more prior terms from the sequence. As such, we must specify initial conditinos such that the sequence can be calculated (think basecase).
+
+Note: The relation is an equation but the result and necessary information to find the next value is a sequence.
+
+Ex:
+
+$a_n = a_{n-1} + 2a_{n-2}$ for $n \geq 2$ where (basecase) $a_0 =2$ and $a_1 =5$.
diff --git a/Sequence.md b/Sequence.md
@@ -0,0 +1,26 @@
+:discrete:
+# Sequence
+
+U2.4.1
+
+## Notes
+
+**Definition:** Sequences are ordered lists mapped to by the integers.
+
+To define a sequence we can use the following notation where n is some arbitrary element:
+
+$a_n = 2n$
+
+This defines the mapping from the integers to the set of all even numbers.
+
+#### Arithmetic Sequence
+
+An arithmetic sequence is a sequence where we start from some constant and then add d times the current value.
+
+This can be explicitly stated as $a_n = a + dn$ where d is some constant.
+
+#### Geometric Sequence
+
+In a geometric sequence we multiply the initial term by the common ratio, defined as r, to the nth power.
+
+A geometric sequence can be stated as $a_n=ar^n$ where r is some constant and n is the iteration, as always.
