1 Vector Spaces
... Definition 1. A vector space (or a linear space) X over a field F (the elements of F are called scalars) is a set of elements called vectors equipped with two (binary) operations, namely vector addition (the sum of two vectors x, y ∈ X is denoted by x + y) and scalar multiplication (the scalar produ ...
... Definition 1. A vector space (or a linear space) X over a field F (the elements of F are called scalars) is a set of elements called vectors equipped with two (binary) operations, namely vector addition (the sum of two vectors x, y ∈ X is denoted by x + y) and scalar multiplication (the scalar produ ...
Math 51H LINEAR SUBSPACES, BASES, AND DIMENSIONS
... Definition. A non-empty subset V of Rn is called a linear subspace if and only if it is closed under addition and under scalar multiplication, i.e., if and only if A, B ∈ V =⇒ A + B ∈ V A ∈ V and c ∈ R =⇒ cA ∈ V Remark. If V is a subspace, then any linear combination of vectors in V must also be in ...
... Definition. A non-empty subset V of Rn is called a linear subspace if and only if it is closed under addition and under scalar multiplication, i.e., if and only if A, B ∈ V =⇒ A + B ∈ V A ∈ V and c ∈ R =⇒ cA ∈ V Remark. If V is a subspace, then any linear combination of vectors in V must also be in ...
Section 2.1,2.2,2.4 rev1
... • We ‘resolve’ vectors into components using the x and y axes system. • Each component of the vector is shown as a magnitude and a direction. • The directions are based on the x and y axes. We use the “unit vectors” i and j to designate the x and y axes. ...
... • We ‘resolve’ vectors into components using the x and y axes system. • Each component of the vector is shown as a magnitude and a direction. • The directions are based on the x and y axes. We use the “unit vectors” i and j to designate the x and y axes. ...
Lecture 3
... A vector v that is a solution to such an equation is called an eigenvector and the number l is called an eigenvalue. Note that v has to be non-zero but l can be zero. Operators in R2 can have from zero to an infinite number of eigenvectors. Let’s look at rotation first. The general rotation operator ...
... A vector v that is a solution to such an equation is called an eigenvector and the number l is called an eigenvalue. Note that v has to be non-zero but l can be zero. Operators in R2 can have from zero to an infinite number of eigenvectors. Let’s look at rotation first. The general rotation operator ...
Vector Algebra and Geometry Scalar and Vector Quantities A scalar
... Let P~Q0 = ka and P~R0 = lb, then P~S 0 = a + b and P~S 0 = ka + lb. By the parallelogram rule we can see that S, S 0 lie in the plane determined by P QR. So if c is a vector whose direction is not parallel to this plane then c will not be of the form ka + lb. c or any multiple of it cannot be expre ...
... Let P~Q0 = ka and P~R0 = lb, then P~S 0 = a + b and P~S 0 = ka + lb. By the parallelogram rule we can see that S, S 0 lie in the plane determined by P QR. So if c is a vector whose direction is not parallel to this plane then c will not be of the form ka + lb. c or any multiple of it cannot be expre ...
24. Orthogonal Complements and Gram-Schmidt
... This is called an orthogonal decomposition because we have decomposed v into a sum of orthogonal vectors. It is significant that we wrote this decomposition with u in mind; v k is parallel to u. If u, v are linearly independent vectors in R3 , then the set {u, v ⊥ , u×v ⊥ } would be an orthogonal ba ...
... This is called an orthogonal decomposition because we have decomposed v into a sum of orthogonal vectors. It is significant that we wrote this decomposition with u in mind; v k is parallel to u. If u, v are linearly independent vectors in R3 , then the set {u, v ⊥ , u×v ⊥ } would be an orthogonal ba ...
Chapter 7
... A vector involves 2 measures, one of direction and one of magnitude. We can represent vector quantities with a directed line segment in which the length represents the magnitude and the angle represents the direction. Vectors, like line segments, can be written using 2 capital letters or a single lo ...
... A vector involves 2 measures, one of direction and one of magnitude. We can represent vector quantities with a directed line segment in which the length represents the magnitude and the angle represents the direction. Vectors, like line segments, can be written using 2 capital letters or a single lo ...
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.)