Matrix Algebra
... The sum of A and B is defined only if the two matrices have the same order which is m × n; in which case they are said to be conformable with respect to addition. Notice that the notation reveals that the matrices are conformable by giving the same indices i = 1, . . . , m and j = 1, . . . , n to the ...
... The sum of A and B is defined only if the two matrices have the same order which is m × n; in which case they are said to be conformable with respect to addition. Notice that the notation reveals that the matrices are conformable by giving the same indices i = 1, . . . , m and j = 1, . . . , n to the ...
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 ...
Vectors and Vector Spaces
... Example 1.1.3. More subspaces of R3 . There are two other important methods to construct subspaces of R3 . Besides the set builder notation used above, we have just considered the method of spanning sets. For example, let S = {v1 , v2 } ⊂ R3 . Then S (S) is a subspace of R3 . Similarly, if T = {v1 } ...
... Example 1.1.3. More subspaces of R3 . There are two other important methods to construct subspaces of R3 . Besides the set builder notation used above, we have just considered the method of spanning sets. For example, let S = {v1 , v2 } ⊂ R3 . Then S (S) is a subspace of R3 . Similarly, if T = {v1 } ...
1. General Vector Spaces 1.1. Vector space axioms. Definition 1.1
... we call the set {a1 , ..., an } the coordinates of w with respect to B. Further, the k-tuple of coordinates [w]B := (a1 , ..., an )TB is referred to as the coordinate matrix of w with respect to B. Theorem 3.11. Suppose B and B 0 = {v10 , ..., vn0 } are two bases for Rn , and w ∈ Rn . Then the relat ...
... we call the set {a1 , ..., an } the coordinates of w with respect to B. Further, the k-tuple of coordinates [w]B := (a1 , ..., an )TB is referred to as the coordinate matrix of w with respect to B. Theorem 3.11. Suppose B and B 0 = {v10 , ..., vn0 } are two bases for Rn , and w ∈ Rn . Then the relat ...
Chapter 12: Three Dimensions
... In three dimensions, vectors are still quantities consisting of a magnitude and a direction, but of course there are many more possible directions. It’s not clear how we might represent the direction explicitly, but the coordinate version of vectors makes just as much sense in three dimensions as in ...
... In three dimensions, vectors are still quantities consisting of a magnitude and a direction, but of course there are many more possible directions. It’s not clear how we might represent the direction explicitly, but the coordinate version of vectors makes just as much sense in three dimensions as in ...
Daily Agenda - math.miami.edu
... does the answer depend on the parameter c? • Compute the intersection of the circles (x − 2)2 + (y − 3)2 = 22 and x2 + y 2 = r2 . How does the answer depend on the parameter r? • What is the distance between two points (x1 , y1 , z1 ) and (x2 , y2 , z2 ) in “Cartesian space” R3 ? Why? (This is a bit ...
... does the answer depend on the parameter c? • Compute the intersection of the circles (x − 2)2 + (y − 3)2 = 22 and x2 + y 2 = r2 . How does the answer depend on the parameter r? • What is the distance between two points (x1 , y1 , z1 ) and (x2 , y2 , z2 ) in “Cartesian space” R3 ? Why? (This is a bit ...
CG-Basics-01-Math - KDD
... Ray / object intersection equations See A.3.5, FVD CIS 636/736: (Introduction to) Computer Graphics ...
... Ray / object intersection equations See A.3.5, FVD CIS 636/736: (Introduction to) Computer Graphics ...
+ v
... If S = {v1, v2, …, vn} is a basis for a vector space V, and v = c1v1 + c2v2 + ··· + cnvn is the expression for a vector v in terms of the basis S, then the scalars c1, c2, …, cn, are called the coordinates of v relative to the basis S. The vector (c1, c2, …, cn) in Rn constructed from these coordina ...
... If S = {v1, v2, …, vn} is a basis for a vector space V, and v = c1v1 + c2v2 + ··· + cnvn is the expression for a vector v in terms of the basis S, then the scalars c1, c2, …, cn, are called the coordinates of v relative to the basis S. The vector (c1, c2, …, cn) in Rn constructed from these coordina ...
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.)