Lecture notes for Linear Algebra
... One usually first encounters a vector depicted as a directed line segment in Euclidean space, or what amounts to the same thing, as an ordered n-tuple of numbers of the form {ai }i=1,...,n . The former is a geometrical description of a vector, whereas the latter can be interpreted as an algebraic re ...
... One usually first encounters a vector depicted as a directed line segment in Euclidean space, or what amounts to the same thing, as an ordered n-tuple of numbers of the form {ai }i=1,...,n . The former is a geometrical description of a vector, whereas the latter can be interpreted as an algebraic re ...
Vector Space
... Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written w ...
... Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written w ...
Slide 1
... Proof: If some vj in S equals a linear combination of the other vectors, then vj can be subtracted from both sides of the equation, producing a linear dependence relation with a nonzero weight ( 1) on vj. [For instance, if v1 c2 v2 c3 v3 , then 0 (1)v1 c2 v2 c3 v3 0v 4 ... 0v p ...
... Proof: If some vj in S equals a linear combination of the other vectors, then vj can be subtracted from both sides of the equation, producing a linear dependence relation with a nonzero weight ( 1) on vj. [For instance, if v1 c2 v2 c3 v3 , then 0 (1)v1 c2 v2 c3 v3 0v 4 ... 0v p ...
Euclidean Space - Will Rosenbaum
... Thus, cj = 0 for all j, as desired. Corollary 22. Orthonormal sets not containing 0 are linearly independent. Corollary 23. An orthogonal subset S ⊆ Rn contains at most n nonzero vectors. ...
... Thus, cj = 0 for all j, as desired. Corollary 22. Orthonormal sets not containing 0 are linearly independent. Corollary 23. An orthogonal subset S ⊆ Rn contains at most n nonzero vectors. ...
The Biquaternions
... • The formula for the product of two biquaternions is the same as for quaternions: (a,b)(c,d) = (ac-db*, a*d+cb) where a, b, c, d C. •Closed •Associative •NOT Commutative •Identity: ...
... • The formula for the product of two biquaternions is the same as for quaternions: (a,b)(c,d) = (ac-db*, a*d+cb) where a, b, c, d C. •Closed •Associative •NOT Commutative •Identity: ...
Geometric Algebra: An Introduction with Applications in Euclidean
... well-suited to the needs of physicists. He distinguished the difference between the dot and cross products of two vectors. Gibbs’s lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students and were later adapted by Edwin Bidwell Wilson into a textbook, Vect ...
... well-suited to the needs of physicists. He distinguished the difference between the dot and cross products of two vectors. Gibbs’s lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students and were later adapted by Edwin Bidwell Wilson into a textbook, Vect ...
1 Introduction Math 120 – Basic Linear Algebra I
... 5. 1 · ~v is by definition the vector of length |1| · ||~v || = ||~v ||, and the two vectors are by definition of the scalar product are collinear and since 1 > 0 it has the same direction of ~v , therefore a parallel vector of equal length, which by definition is equal to ~v . 6. We need to show th ...
... 5. 1 · ~v is by definition the vector of length |1| · ||~v || = ||~v ||, and the two vectors are by definition of the scalar product are collinear and since 1 > 0 it has the same direction of ~v , therefore a parallel vector of equal length, which by definition is equal to ~v . 6. We need to show th ...
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.