commit 6157740b581836c5c0aa26f213f6d0245a407f32
parent e797a1685fc9250664a8f1700a4bda33c6f99d60
Author: Andrew <andrewlaack1@gmail.com>
Date: Thu, 4 Jul 2024 12:20:13 -0500
Took notes on closure and subspaces
Diffstat:
3 files changed, 34 insertions(+), 0 deletions(-)
diff --git a/Closure.md b/Closure.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Closure
+
+Khan
+
+## Notes
+
+**Definition:** Closure means that performing some arbitrary operation (pick one, but not necessarily all) on any member of a set will result in another element of a set.
+
+In the context of subspaces, we have closure under scalar multiplication and vector addition because these operations on any element of the [[LinearSubspace.md]] set results in another element of the set (by definition).
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -19,3 +19,5 @@ The basis of linear algebra is solving systems of equations.
[[UnitVector.md]]
[[Span.md]]
[[LinearIndependence.md]]
+[[LinearSubspace.md]]
+[[Closure.md]]
diff --git a/LinearSubspace.md b/LinearSubspace.md
@@ -0,0 +1,22 @@
+:lin-alg:
+# Linear Subspace (or simply subspace)
+
+Khan
+
+## Notes
+
+**Definition:** A linear subspace is a subset (inclusive of the subset being the entire set) of a space of equal or greater cardinality where the linear subspace contains the zero vector.
+
+Things like a plane that passes through the origin in R^n, a line that passes through the origin in R^n, or R^n itself are all specific linear subspaces (or just subspaces for short).
+
+Additionally, we have closure under addition meaning any two vectors added together in the set are still contained within the subspace and the superspace.
+
+The final thing is it is closed under multiplication. As such, multiplying any vector in the set by any scalar means the result is also in the subspace.
+
+### Summary
+
+Three rules neccesary and sufficient for subspace definition:
+
+1. Closed under multiplication (scalar)
+2. Closed under addition (with other elements)
+3. Contains zero vector
