Dot Product, Cross Product, Determinants
... A2 = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2 . Hence A = k~xk with the vector ~x defined by ~x := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). Note that x j = ak b` − a` bk where 1. j, k, ` are different 2. k, ` are in “positive order” if we arrange 1,2,3 on a circle. This vec ...
... A2 = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2 . Hence A = k~xk with the vector ~x defined by ~x := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). Note that x j = ak b` − a` bk where 1. j, k, ` are different 2. k, ` are in “positive order” if we arrange 1,2,3 on a circle. This vec ...
Notes on Weak Topologies
... topology generated by the subbasis S is called the product topology and the space X with this topology is called the product space. (9) Let (X, τ ) be a topological space. A collection A = {Ui }i∈I of open subsets of X such that X = ∪i∈I Ui is called an open cover of X. A subcollection B ⊆ A is call ...
... topology generated by the subbasis S is called the product topology and the space X with this topology is called the product space. (9) Let (X, τ ) be a topological space. A collection A = {Ui }i∈I of open subsets of X such that X = ∪i∈I Ui is called an open cover of X. A subcollection B ⊆ A is call ...
Synopsis of Geometric Algebra
... Although the vector space Vn is closed under vector addition, it is not closed under multiplication, as the contraction rule (1.3) shows. Instead, by multiplication and addition the vectors of Vn generate a larger linear space Gn = G(Vn ) called the geometric algebra of Vn . This linear space is, of ...
... Although the vector space Vn is closed under vector addition, it is not closed under multiplication, as the contraction rule (1.3) shows. Instead, by multiplication and addition the vectors of Vn generate a larger linear space Gn = G(Vn ) called the geometric algebra of Vn . This linear space is, of ...
quaternions slides
... Written as (cos(θ/2), v sin(θ/2)) "Unit quaternion": q∙q = 1 (if v is a unit vector) Maintain unit quaternion by normalizing v Why not use s for angle, v for axis directly? ...
... Written as (cos(θ/2), v sin(θ/2)) "Unit quaternion": q∙q = 1 (if v is a unit vector) Maintain unit quaternion by normalizing v Why not use s for angle, v for axis directly? ...
Matrix Analysis
... appearing in the third row and first column. A matrix A may also be denoted as [aij], where aij denotes the general element of A appearing in the ith row and jth column. A matrix having r rows and с columns has order (or size) "r by c," usually written rc. Two matrices are equal if they have the sa ...
... appearing in the third row and first column. A matrix A may also be denoted as [aij], where aij denotes the general element of A appearing in the ith row and jth column. A matrix having r rows and с columns has order (or size) "r by c," usually written rc. Two matrices are equal if they have the sa ...
Topology Proceedings - Topology Research Group
... Proposition 2. Let Z == ZI X Z2 be the product of discrete metric spaces with the 11 metric. Assume that ZI and Z2 have property A. Then Z has property A. Proposition 3. Let W C Z and let Z have property A. Then W has property A. Proof: Let an : Z --t P( Z) be a sequence of maps from the definition ...
... Proposition 2. Let Z == ZI X Z2 be the product of discrete metric spaces with the 11 metric. Assume that ZI and Z2 have property A. Then Z has property A. Proposition 3. Let W C Z and let Z have property A. Then W has property A. Proof: Let an : Z --t P( Z) be a sequence of maps from the definition ...