Vector Spaces and Linear Transformations
... If H is a subspace of V , then H is closed for the addition and scalar multiplication of V , i.e., for any u, v ∈ H and scalar c ∈ R, we have u + v ∈ H, cv ∈ H. For a nonempty set S of a vector space V , to verify whether S is a subspace of V , it is required to check (1) whether the addition and s ...
... If H is a subspace of V , then H is closed for the addition and scalar multiplication of V , i.e., for any u, v ∈ H and scalar c ∈ R, we have u + v ∈ H, cv ∈ H. For a nonempty set S of a vector space V , to verify whether S is a subspace of V , it is required to check (1) whether the addition and s ...
Complex vectors
... Complex vectors are vectors whose components can be complex numbers. They were introduced by the famous American physicist J. WILLARD GlBBS, sometimes called the 'Maxwell of America', at about the same period in the 1880's as the real vector algebra, in a privately printed but widely circulated pamp ...
... Complex vectors are vectors whose components can be complex numbers. They were introduced by the famous American physicist J. WILLARD GlBBS, sometimes called the 'Maxwell of America', at about the same period in the 1880's as the real vector algebra, in a privately printed but widely circulated pamp ...
classwork geometry 5/13/2012
... In stereometry, hyperbola can be defined as a conic cross-section, similar to parabola and ellipse. Namely, hyperbola is the curve of intersection between a right circular conical surface and a plane that cuts through both halves of the cone. For the other major types of conic sections, the ellipse ...
... In stereometry, hyperbola can be defined as a conic cross-section, similar to parabola and ellipse. Namely, hyperbola is the curve of intersection between a right circular conical surface and a plane that cuts through both halves of the cone. For the other major types of conic sections, the ellipse ...
CMSC 425: Lecture 6 Affine Transformations and Rotations
... as the inventor of graph theory. Among his many theorems is one that states that the composition any number of rotations in three-space can be expressed as a single rotation in 3-space about an appropriately chosen vector. Euler also showed that any rotation in 3-space could be broken down into exac ...
... as the inventor of graph theory. Among his many theorems is one that states that the composition any number of rotations in three-space can be expressed as a single rotation in 3-space about an appropriately chosen vector. Euler also showed that any rotation in 3-space could be broken down into exac ...
DRAFT Errors will be corrected before printing. Final book will be...
... We have shown that if we take any two forces that act at the same point, acting at an angle of u to each other, the forces may be composed to obtain the resultant of these two forces. Furthermore, the resultant of any two forces is unique because there is only one parallelogram that can be formed wi ...
... We have shown that if we take any two forces that act at the same point, acting at an angle of u to each other, the forces may be composed to obtain the resultant of these two forces. Furthermore, the resultant of any two forces is unique because there is only one parallelogram that can be formed wi ...
11.6 Dot Product and the Angle between Two Vectors
... Let v, w, and a be nonzero vectors such that v ! a D w ! a. Is it true that v D w? Either prove this or give a counterexThe equality v ! a D w ! a is equivalent to the following equality: v!aD w!a v!a"w!aD 0 .v " w/ ! a D 0 ...
... Let v, w, and a be nonzero vectors such that v ! a D w ! a. Is it true that v D w? Either prove this or give a counterexThe equality v ! a D w ! a is equivalent to the following equality: v!aD w!a v!a"w!aD 0 .v " w/ ! a D 0 ...
Octave Tutorial 2
... To extract a subset of elements, you can use the colon : operator. For example, the command octave#:#> X(2,1:3) extracts, from the second row, all the elements between the first and the third column (included). Try it! To extract an entire row or column, use the colon : operator like this, octave#:# ...
... To extract a subset of elements, you can use the colon : operator. For example, the command octave#:#> X(2,1:3) extracts, from the second row, all the elements between the first and the third column (included). Try it! To extract an entire row or column, use the colon : operator like this, octave#:# ...
Quaternions and isometries
... by a reflection across the line ∩ 2 ; thus we see that R2 R1 is a rotation of R3 about the axis L of an angle equal to twice the angle between the planes j . This leads us to the following definition of a rotation. Definition 6.1.3 A rotation of R3 is the composition of reflections across two disti ...
... by a reflection across the line ∩ 2 ; thus we see that R2 R1 is a rotation of R3 about the axis L of an angle equal to twice the angle between the planes j . This leads us to the following definition of a rotation. Definition 6.1.3 A rotation of R3 is the composition of reflections across two disti ...
![arXiv:math/0403252v1 [math.HO] 16 Mar 2004](http://s1.studyres.com/store/data/017716978_1-976b7d80197f8d34f66860694a7698bb-300x300.png)
Review of Matrices and Vectors
... and only if their inner product is zero. Two vectors are orthonormal if, in addition to being orthogonal, the length of each vector is 1. From the concepts just discussed, we see that an arbitrary vector a is turned into a vector an of unit length by performing the operation an = a/||a||. Clearly, t ...
... and only if their inner product is zero. Two vectors are orthonormal if, in addition to being orthogonal, the length of each vector is 1. From the concepts just discussed, we see that an arbitrary vector a is turned into a vector an of unit length by performing the operation an = a/||a||. Clearly, t ...
Homework 2. Solutions 1 a) Show that (x, y) = x1y1 + x2y2 + x3y3
... (x ) + (x2 )2 + (x3 )2 = 0, then x1 = x2 = x3 = 0, i.e. x = 0. This we proved positive-definiteness. All conditions are checked. Hence B(x, y) = x1 y 1 + x2 y 2 + x3 y 3 is indeed a scalar product in R3 Remark Note that x1 , x2 , x3 —are components of the vector, do not be confused with exponents! S ...
... (x ) + (x2 )2 + (x3 )2 = 0, then x1 = x2 = x3 = 0, i.e. x = 0. This we proved positive-definiteness. All conditions are checked. Hence B(x, y) = x1 y 1 + x2 y 2 + x3 y 3 is indeed a scalar product in R3 Remark Note that x1 , x2 , x3 —are components of the vector, do not be confused with exponents! S ...
1 Basis
... The determinant of this system 6= 0, so this system has a solution for each a and b. Example. Different sequences of vectors can span the same sets. For example R2 is spanned by each of the following sequences: (a) v1 = (1, 0), v2 = (0, 1); (b) v1 = (−1, 0), v2 = (0, 1); (c) v1 = (1, 1), v2 = (0, 1) ...
... The determinant of this system 6= 0, so this system has a solution for each a and b. Example. Different sequences of vectors can span the same sets. For example R2 is spanned by each of the following sequences: (a) v1 = (1, 0), v2 = (0, 1); (b) v1 = (−1, 0), v2 = (0, 1); (c) v1 = (1, 1), v2 = (0, 1) ...
Cross product
In mathematics and vector calculus, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R3) and is denoted by the symbol ×. The cross product a × b of two linearly independent vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product).If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e. a × b = −b × a) and is distributive over addition (i.e. a × (b + c) = a × b + a × c). The space R3 together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or ""handedness"". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, then only the 3-dimensional cross product qualifies. (See § Generalizations, below, for other dimensions.)