
1. New Algebraic Tools for Classical Geometry
... Any degenerate algebra can be embedded in a non-degenerate algebra of larger dimension, and it is almost always a good idea to do so. Otherwise, there will be subspaces without a complete basis of dual vectors, which will complicate algebraic manipulations. The n-dimensional vector spaces of every p ...
... Any degenerate algebra can be embedded in a non-degenerate algebra of larger dimension, and it is almost always a good idea to do so. Otherwise, there will be subspaces without a complete basis of dual vectors, which will complicate algebraic manipulations. The n-dimensional vector spaces of every p ...
L3 Vector Operations
... In an xy-coordinate system, the direction of a nonzero vector v is determined by the angles α and β between v and the unit vectors i and j, and in an xyzcoordinate system, the direction is determined by the angles α, β, and γ between v and the unit vectors i, j, and k. In both 2- space and 3-space, ...
... In an xy-coordinate system, the direction of a nonzero vector v is determined by the angles α and β between v and the unit vectors i and j, and in an xyzcoordinate system, the direction is determined by the angles α, β, and γ between v and the unit vectors i, j, and k. In both 2- space and 3-space, ...
CURRICULUM SUMMARY – September to October 2008
... CURRICULUM SUMMARY – September to December 2016 ...
... CURRICULUM SUMMARY – September to December 2016 ...
17. Inner product spaces Definition 17.1. Let V be a real vector
... hu, vi ≤ kuk · kvk. Definition 17.6. Let V be a real vector space with an inner product. We say that two vectors v and w are orthogonal if hu, vi = 0. We say that a basis v1 , v2 , . . . , vn is an orthogonal basis if the vectors v1 , v2 , . . . , vn are pairwise orthogonal. If in addition the vecto ...
... hu, vi ≤ kuk · kvk. Definition 17.6. Let V be a real vector space with an inner product. We say that two vectors v and w are orthogonal if hu, vi = 0. We say that a basis v1 , v2 , . . . , vn is an orthogonal basis if the vectors v1 , v2 , . . . , vn are pairwise orthogonal. If in addition the vecto ...
Lines and Planes Linear algebra is the study of linearity in its most
... scalar multiple of such a vector points in the direction of L, and all such vectors obey the distributive property of scalar multiplication over vector addition. Therefore, the set of scalar multiples of Q – P forms a vector space. It follows that all every vector X whose tip lies on L has the form ...
... scalar multiple of such a vector points in the direction of L, and all such vectors obey the distributive property of scalar multiplication over vector addition. Therefore, the set of scalar multiples of Q – P forms a vector space. It follows that all every vector X whose tip lies on L has the form ...
Linear Vector Space
... 10 Axioms of a Vector Space 1. u + v is in V (closure under addition) 2. u + v = v + u (commutativity) 3. (u + v) + w = u + (v + w) (associativity) 4. There exists an element 0 in V , called a zero vector, such that u+0=u. 5. For each u in V , there is an element −u in V such that u + (−u)=0. 6. cu ...
... 10 Axioms of a Vector Space 1. u + v is in V (closure under addition) 2. u + v = v + u (commutativity) 3. (u + v) + w = u + (v + w) (associativity) 4. There exists an element 0 in V , called a zero vector, such that u+0=u. 5. For each u in V , there is an element −u in V such that u + (−u)=0. 6. cu ...
Linear Algebra
... The inverse (A-1) is defined such at A A-1 is I. Not every matrix has an inverse. If no inverse exists, then the matrix is called singular (non invertible) If A is nonsingular, so is A-1 If A, B are nonsingular, then AB is also non singular and (AB) -1 = B -1A -1 (Note reversed order.) If ...
... The inverse (A-1) is defined such at A A-1 is I. Not every matrix has an inverse. If no inverse exists, then the matrix is called singular (non invertible) If A is nonsingular, so is A-1 If A, B are nonsingular, then AB is also non singular and (AB) -1 = B -1A -1 (Note reversed order.) If ...
ppt - Geometric Algebra
... • We carry round the blade and coefficient together (in a tuple) • We have a geometric product and a projection operator • The geometric product works on the individual blades • Ideally, do not multiply coefficients when result is not needed • All expressed in a functional programming language ...
... • We carry round the blade and coefficient together (in a tuple) • We have a geometric product and a projection operator • The geometric product works on the individual blades • Ideally, do not multiply coefficients when result is not needed • All expressed in a functional programming language ...
ANALYTICAL MATHEMATICS
... Analytical Mathematics is a course designed for students who have successfully completed the Algebra II With Trigonometry course. It is considered to be parallel in rigor to Precalculus. This course provides a structured introduction to important areas of emphasis in most postsecondary studies that ...
... Analytical Mathematics is a course designed for students who have successfully completed the Algebra II With Trigonometry course. It is considered to be parallel in rigor to Precalculus. This course provides a structured introduction to important areas of emphasis in most postsecondary studies that ...
5.1 - shilepsky.net
... We discuss an algebraic structure that is an abstraction of vectors in R n. Definition: Let V be a set of objects with two operations defined on the set called addition and scalar multiplication. That is, for each u and v in V, there is an element u+v in V called the sum, and for each u in V and sca ...
... We discuss an algebraic structure that is an abstraction of vectors in R n. Definition: Let V be a set of objects with two operations defined on the set called addition and scalar multiplication. That is, for each u and v in V, there is an element u+v in V called the sum, and for each u in V and sca ...
02 Dot Product
... There are special unit vectors that point in the coordinate directions. The vector i points in the x-direction, j in the y-direction, and k in the z-direction. (The names change in more than three dimensions, so we won’t run out of letters!) Now let’s explore what happens when a dot product is zero ...
... There are special unit vectors that point in the coordinate directions. The vector i points in the x-direction, j in the y-direction, and k in the z-direction. (The names change in more than three dimensions, so we won’t run out of letters!) Now let’s explore what happens when a dot product is zero ...
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered an order zero quantity, and a vector is an order one quantity, then a bivector can be thought of as being of order two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities.Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product a ∧ b is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product is antisymmetric, so b ∧ a is the negation of the bivector a ∧ b, producing the opposite orientation, and a ∧ a is the zero bivector.Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. The bivector a ∧ b has a magnitude equal to the area of the parallelogram with edges a and b, has the attitude of the plane spanned by a and b, and has orientation being the sense of the rotation that would align a with b.