Finite-Dimensional Vector Spaces
... The list B (of length n) from step j −1 spans V , and thus adjoining any vector to it produces a linearly dependent list. In particular, the list of length (n + 1) obtained by adjoining uj to B, placing it just after u1 , . . . , uj−1 , is linearly dependent. By the linear dependence lemma (2.4), on ...
... The list B (of length n) from step j −1 spans V , and thus adjoining any vector to it produces a linearly dependent list. In particular, the list of length (n + 1) obtained by adjoining uj to B, placing it just after u1 , . . . , uj−1 , is linearly dependent. By the linear dependence lemma (2.4), on ...
11-15-16 Matrices Multiplication
... main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. ...
... main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. ...
NOTES ON LINEAR ALGEBRA
... We now define the transpose of a matrix. For us, the main use will be in studying symmetric matrices, matrices that are equal to their transpose. We write AT for the transpose of the matrix A, and we form AT as follows: the first row of A becomes the first column of AT; the second row of A becomes t ...
... We now define the transpose of a matrix. For us, the main use will be in studying symmetric matrices, matrices that are equal to their transpose. We write AT for the transpose of the matrix A, and we form AT as follows: the first row of A becomes the first column of AT; the second row of A becomes t ...
Chapter 1 Linear Algebra
... A related notion to a vector space is that of a vector subspace. Suppose that V is a vector space and let W ⊂ V be a subset. This means that W consists of some (but not necessarily all) of the vectors in V. Since V is a vector space, we know that we can add vectors in W and multiply them by scalars, ...
... A related notion to a vector space is that of a vector subspace. Suppose that V is a vector space and let W ⊂ V be a subset. This means that W consists of some (but not necessarily all) of the vectors in V. Since V is a vector space, we know that we can add vectors in W and multiply them by scalars, ...
On the Associative Nijenhuis Relation
... The quasi-shuffle product ∗ essentially embodies the structure of the Rota-Baxter relation (3). The case of weight λ = 0 gives the “trivial” Rota-Baxter algebra, i.e. relation (3) without the second term on the left-hand side. This construction was essentially given in [14] using a non-recursive not ...
... The quasi-shuffle product ∗ essentially embodies the structure of the Rota-Baxter relation (3). The case of weight λ = 0 gives the “trivial” Rota-Baxter algebra, i.e. relation (3) without the second term on the left-hand side. This construction was essentially given in [14] using a non-recursive not ...
4-2
... A square matrix is any matrix that has the same number of rows as columns; it is an n × n matrix. The main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. ...
... A square matrix is any matrix that has the same number of rows as columns; it is an n × n matrix. The main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. ...
4 Images, Kernels, and Subspaces
... (Solution) Let m be the smallest integer so that Am = 0, as in Exercise 78. According to that exercise, we may choose v so that the vectors v, Av, A2 v, . . . , Am−1 v are linearly independent. We know that any collection of more than n vectors in Rn is linearly dependent, so this collection may hav ...
... (Solution) Let m be the smallest integer so that Am = 0, as in Exercise 78. According to that exercise, we may choose v so that the vectors v, Av, A2 v, . . . , Am−1 v are linearly independent. We know that any collection of more than n vectors in Rn is linearly dependent, so this collection may hav ...
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.)