commit 948b8b94ed8f2dc70a04a1634a403fcdb060ec2a
parent 6157740b581836c5c0aa26f213f6d0245a407f32
Author: Andrew <andrewlaack1@gmail.com>
Date: Fri, 5 Jul 2024 12:13:52 -0500
Took some la notes
Diffstat:
7 files changed, 117 insertions(+), 0 deletions(-)
diff --git a/AngleBetweenVectors.md b/AngleBetweenVectors.md
@@ -0,0 +1,17 @@
+:lin-alg:
+# Angle Between Vectors
+
+Khan
+
+## Notes
+
+**Definition:** The angle between two vectors is the angle between the two vectors when their tails are positioned at the zero vector.
+
+## Calculation
+
+1. Find magnitude of both vectors ([[DistanceCalculation.md]])
+2. Take dot product divided by lengths of vectors to find cosine of the angle (solve)
+ - cos(theta) = (u dot v)/(||u||||v||)
+
+
+## Usage
diff --git a/BasisOfSubspace.md b/BasisOfSubspace.md
@@ -0,0 +1,14 @@
+:lin-alg:
+# Basis of a Subspace
+
+Khan
+
+## Notes
+
+**Definition:** The basis of a subspace is the span of all linearly independent vectors of a given set of vectors.
+
+Basically, we have a set of vectors and this is the basis of a subspace whereby the subspace is the span of the vectors (all scalar and additive combinations).
+
+Note:
+
+The basis is the **minimum** set of vectors to describe the span of the subspace. As such, all vectors must be linearly independent.
diff --git a/CrossProduct.md b/CrossProduct.md
@@ -0,0 +1,21 @@
+:lin-alg:
+# Cross Product
+
+Khan
+
+## Notes
+
+**Definition:** The cross product of two vectors is the vector orthogonal to them.
+
+The cross product is only defined in R^3.
+
+### Calculation
+
+To calculate the cross product we simply do the following:
+
+a = [a_1, a_2, a_3]
+b = [b_1, b_2, b_3]
+
+o = [a_2b_3 - a_3b_2 , a_3b_1 - a_1b_3 , a_1b_2 - a_2b_1]
+
+Conceptually, we take the determinant of the bottom two rows then reverse determinent of top and bottom rows then deteminent of top two rows.
diff --git a/DotProduct.md b/DotProduct.md
@@ -10,3 +10,9 @@ CS331 + Linear Algebra
This can be visualized as the length of one vector, v, projected onto another vector, y, multiplied by the length of the vector y. Additionally, if two vectors generally have a different direction, their dot product is negative. This is why the on same side of plane algorithm works (see cs331 code), because if two vectors are on the same side of the normal vector of a plane, then they will both have negative or positive dot products.
This value is zero if the vectors are orthogonal.
+
+### Additional Thoughts
+
+The dotproduct in a geometric sense is u dot v = ||u|| ||v|| cos(theta) where theta is the angle between the two vectors. As such, when the angle between them is greater than 90 and less than 270 we find the dot product is negative.
+
+This intuition leads to how we can find the angle between two vectors namely theta above.
diff --git a/EquationOfAPlane.md b/EquationOfAPlane.md
@@ -0,0 +1,44 @@
+:lin-alg:
+# Equation of a Plane
+
+Khan
+
+## Notes
+
+**Definition:** The equation of a plane is the equation that defines all points on the plane as a combination of n variables where n is the number of dimensions we are in. This is the definition of plane when in 3d space and a hyperplane in higher dimensions.
+
+### Plane Formula
+
+The general formula for a 3d plane is as follows:
+
+ax + by + cz = d
+
+where a,b,c, and d are coeficcients and x,y, and z are variables.
+
+### Hyperplane formula
+
+A hyperplane in n dimensional space is defined as follows:
+
+a_1 x_1 + a_2 x_2 + ... + a_n x_n = d
+
+Where, again, all a sub values are coeficcients along with d and all x values are variables.
+
+### Calculation
+
+When we have a normal vector and a point this is very simple.
+
+Steps:
+
+1. Plug in normal vector for equation of the plane:
+ - ax + by + cz = d where a,b,c are the x,y, and z axis components of the vector
+2. Plug in representative point as x,y, and z and then solve for d
+
+This can be extrapolated into higher dimensions for hyper-planes assuming we still have a representative point and the normal vector.
+
+### No Normal Vector
+
+To find the normal vector of a plane when we just have three points or two vectors and one point we can find the normal vector as follows:
+
+1. If we have three points then find two vectors on the plane by taking the difference between a reference point and the two other points on the plane. If these vectors are colinear (dependent) then we don't have enough information to get the formula for the plane.
+2. Take the cross product of both vectors to find the normal vector
+3. Complete calculation steps above
diff --git a/LawOfCosines.md b/LawOfCosines.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Law of Cosines
+
+SS
+
+## Notes
+
+**Definition:** The law of cosines is defined as c^2 = a^2 + b^2 - 2ab cos(C) where a and b are side lengths and c is the side length to be found that is opposite of the angle C.
+
+When using the law of cosines it is important to note we are finding the side length opposite of the known angle.
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -21,3 +21,8 @@ The basis of linear algebra is solving systems of equations.
[[LinearIndependence.md]]
[[LinearSubspace.md]]
[[Closure.md]]
+[[BasisOfSubspace.md]]
+[[AngleBetweenVectors.md]]
+[[LawOfCosines.md]]
+[[EquationOfAPlane.md]]
+[[CrossProduct.md]]
