commit 8336cb0ab6c3e5ac0144d96d5083a4f3dc866bfc
parent c99525717defdf2f1d43da9e66b4b57edc65fd93
Author: Andrew <andrewlaack1@gmail.com>
Date: Wed, 28 Aug 2024 18:21:41 -0500
Took notes on linear algebra'
Diffstat:
9 files changed, 61 insertions(+), 8 deletions(-)
diff --git a/ComplexVectorSpace.md b/ComplexVectorSpace.md
@@ -0,0 +1,8 @@
+:lin-alg:
+# Complex Vector Space
+
+Ch 1
+
+## Notes
+
+**Definition:** A complex vector space is a vector space on the complex numbers (C).
diff --git a/DirectSum.md b/DirectSum.md
@@ -0,0 +1,8 @@
+:lin-alg:
+# Direct Sum
+
+Ch 1
+
+## Notes
+
+**Definition:** A direct sum (denoted with a circle with a plus in it like xor) is a sum of two vector spaces that are disjoint except for the zero vector.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -102,3 +102,4 @@ Unit 2.4 (integers and division):
- [CongruenceClass](CongruenceClass.md)
- [CaesarCipher](CaesarCipher.md)
- [VigenereCipher](VigenereCipher.md)
+ -
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -10,13 +10,13 @@ The basis of linear algebra is solving systems of equations.
Linear Algebra Done Right:
Chapter 1:
- - Vector Space
- - Tuple (list - finite)
- - real vector space
- - complex vector space
- - subspace (also a vector space)
- - sum (of vector space)
- - direct sum
+ - [VectorSpace](VectorSpace.md)
+ - [Tuple](Tuple.md)
+ - [RealVectorSpace](RealVectorSpace.md)
+ - [ComplexVectorSpace](ComplexVectorSpace.md)
+ - [Subspace](Subspace.md)
+ - [SumOfVectorSpaces](SumOfVectorSpaces.md)
+ - [DirectSum](DirectSum.md)
Khan Academy:
diff --git a/RealVectorSpace.md b/RealVectorSpace.md
@@ -0,0 +1,8 @@
+:lin-alg:
+# Real Vector Space
+
+Ch 1
+
+## Notes
+
+**Definition:** A real vector space is a vector space on the real numbers.
diff --git a/Subspace.md b/Subspace.md
@@ -1,10 +1,16 @@
-:ml:
+:ml: :lin-alg:
# Subspace
ML D5
## Notes
+
+#### ML Context
**Definition:** A subspace is a lower dimensional space.
Often we find that many higher dimensional points all reside in or near a similar lower dimensional subspace which is the basis for [[Projection.md]]
+
+#### Lin Alg Context
+
+**Definition:** A subspace is a subset of a vector space.
diff --git a/SumOfVectorSpaces.md b/SumOfVectorSpaces.md
@@ -0,0 +1,8 @@
+:lin-alg:
+# Sum of Vector Spaces
+
+Ch1
+
+## Notes
+
+**Definition:** The sum of two vector spaces is another vector space which is formed by all combinations of vectors in both spaces.
diff --git a/Tuple.md b/Tuple.md
@@ -0,0 +1,12 @@
+:lin-alg:
+# Tuple
+
+Ch 1
+
+## Notes
+
+**Definition:** A tuple is an ordered list of elements (like a set but ordered).
+
+A tuple in 2d is an ordered pair.
+
+Often tuples are used to represent points in space.
diff --git a/VectorSpace.md b/VectorSpace.md
@@ -8,3 +8,5 @@ AM Ch8
**Definition:** A vector space is a set of elements that obey certain properties.
An interesting thing about vector spaces is the union of two vector spaces is itself a vector space.
+
+Vector spaces must include an identity vector (under multiplication and addition) as well as be closed under multiplication and addition. Additionally, they must include additive inverses, respect distributivity, and associativity.
