commit 3b0736b1bf65cee2291c7d5c60dd6ba84b40cca1
parent f62b8dc5520abe3d55bf3d01a24a16259abd7280
Author: Andrew <andrewlaack1@gmail.com>
Date: Fri, 16 Aug 2024 16:28:09 -0500
Completed this stuff
Diffstat:
7 files changed, 77 insertions(+), 0 deletions(-)
diff --git a/Assembly.md b/Assembly.md
@@ -0,0 +1,10 @@
+:asm: :index:
+# Assembly Language
+
+Main Links For Assembly Language
+
+---
+
+Arm Assembly Language 2nd Edition Notes:
+
+
diff --git a/DemorgansLaw.md b/DemorgansLaw.md
@@ -0,0 +1,40 @@
+:discrete:
+# Demorgan's Laws
+
+1.3.2
+
+## Notes
+
+**Definition:** These are two fundamental laws of boolean algebra that can be simply derived.
+
+$\neg (p \wedge q) \equiv \neg p \vee \neg q$
+
+$\neg (p \vee q) \equiv \neg p \wedge \neg q$
+
+#### Showing these laws are true with truth tables
+
+First law (columns 4 and 7 are being shown as equivalent):
+
+| p | q | $p \wedge q$ | $\neg(p \wedge q)$ | $\neg p$ | $\neg q$ | $\neg p \vee \neg q$ |
+| - | - | - | - | - | - | - |
+| T | T | T | F | F | F | F
+| T | F | F | T | F | T | T
+| F | T | F | T | T | F | T
+| F | F | F | T | T | T | T
+
+Second law (columns 4 and 7 are being shown as equivalent):
+
+| p | q | $p \vee q$ | $\neg(p \vee q)$ | $\neg p$ | $\neg q$ | $\neg p \wedge \neg q$ |
+| - | - | - | - | - | - | - |
+| T | T | T | F | F | F | F
+| T | F | T | F | F | T | F
+| F | T | T | F | T | F | F
+| F | F | F | T | T | T | T
+
+#### What these are saying
+
+The first law states that not p and q is the same as not p or not q.
+
+The second law states that not p or q is the same is not p and not q.
+
+This is basically the distributive property of boolean logic whereby we flip the and/or connective and distribute the negation.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -24,3 +24,4 @@ Unit 1.3:
- [[Tautology.md]] - Always true - use the symbol t
- [[Contradiction.md]] - Always false
- [[Contingency.md]] - Must be evaluated
+ - [[DemorgansLaw.md]]
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -72,3 +72,4 @@ Khan Unit 2:
- [[GaussianElimination.md]]
- [[EigenVector.md]]
- [[Transpose.md]]
+ - [[OrthogonalComplement.md]]
diff --git a/OrthogonalComplement.md b/OrthogonalComplement.md
@@ -0,0 +1,14 @@
+:lin-alg:
+# Orthogonal Complement
+
+Khan U3
+
+## Notes
+
+**Definition:** The orthogonal complement of a subspace is the subspace such that the dot product between any vectors (one from each subspace) are 0.
+
+The orthogonal complement of the subspace V in $\R^n$ is defined as follows:
+
+$V^\perp = \{\vec{x} \in \R^n | \vec{x} \cdot \vec{v} = 0 \text{ and } \vec{v} \in V \}$
+
+The orthogonal complement of a subspace is a subspace in all cases as it respects scalar multiplication, vector addition, and contains the zero vector.
diff --git a/Transpose.md b/Transpose.md
@@ -92,3 +92,13 @@ A = [a_1]
[a_3]
A^T = [a_1 a_2 a_3]
+
+#### Column Space
+
+The transpose of a matrix has a column space equivalent to the row space of the original matrix.
+
+C(A^T) = Row space(A)
+
+#### Null Space
+
+The null space of the transpose is called the left nullspace.
diff --git a/index.md b/index.md
@@ -25,6 +25,7 @@ This is the index for my main note classifications. I will maintain this as a ho
[[Calculus.md]]
[[Algorithms.md]]
[[Physics.md]]
+[[Assembly.md]]
## Things to Learn More About
