Vectors scalar multiplication

For simplicity we speak of a mechanism or a reaction, rather than a mechanism vector or reaction vector. The distinction lies in the fact that a reaction r (or mechanism) is essentially the same whether its rate of advancement is p or a, whereas pr and or are different vectors (for p a). Therefore, a reaction could properly be defined as a one-dimensional vector space which contains all the scalar multiples of a single reaction vector, but the mathematical development is simpler if a reaction is defined as a vector. This leaves open the question of when two reactions, or two mechanisms, are essentially different from a chemical viewpoint, which will be taken up... 

Exercise 1.19 Define an arrow in R to be an ordered pair (p, pf), where Pi and p2 are each a triple of real numbers. (Think of pi as the initial point and p as the endpoint.) Define a relation on the set of arrows by (/ b pf) (qt q .) if and only if P2 — Pi = qi qi- Show that this is an equivalence relation. Now think of each arrow as a point in R. Does the usual addition in R survive the equivalence relation If so, is the resulting addition on equivalence classes of arrows the same as the addition of 3-vectors you learned in linear algebra What about scalar multiplication in R . Find an injective and surjective linear function from R / to R. (Hint it will help to introduce some notation for (r, r2, r, r4, r, r(,) e R in the equivalence class corresponding to (si, 52, 53) e R. )... 

In this chapter we introduce complex linear algebra, that is, linear algebra where complex numbers are the scalars for scalar multiplication. This may feel like review, even to readers whose experience is limited to real linear algebra. Indeed, most of the theorems of linear algebra remain true if we replace R by C because the axioms for a real vector space involve only addition and multiplication of real numbers, the definition and basic theorems can be easily adapted to any set of scalars where addition and multiplication are defined and reasonably well behaved, and the complex numbers certainly fit the bill. However, the examples are different. Furthermore, there are theorems (such as Proposition 2.11) in complex linear algebra whose analogues over the reals are false. We will recount but not belabor old theorems, concentrating on new ideas and examples. The reader may find proofs in any number of... 

For example, the real line R is not a complex vector space under the usual multipUcation of real numbers by complex numbers. It is possible for the product of a complex number and a real number to be outside the set of real numbers for instance, (z)(3) = 3i R. So the real line R is not closed under complex scalar multiplication. 

The trivial complex vector space has one element, the zero vector 0. Addition is defined by 0 -I- 0 = 0 for any complex number c, define the scalar multiple of 0 by c to be 0. Then all the criteria of Definition 2.1 are trivially true. For example, to check distributivity, note that for any c e C we have... 

If a subset W of a vector space V satisfies the definition of a vector space, with addition and scalar multiplication defined by the same operation as in V, then W is called a vector subspace or, more succinctly, a subspace of V. For example, the trivial subspace 0 is a subspace of any vector space. 

Because the sum of two continuous functions is continuous, and any scalar multiple of a continuous function is continuous, C[—1, 1] is indeed a vector space. 

The notion of a linear transformation is crucial. A function from a (complex) vector space to a (complex) vector space is a (complex) linear transformation if it preserves addition and (complex) scalar multiplication. Here is a more explicit definition. 

Proof. If A is diagonal, then an easy computation shows that AD — DA = 0. To prove the other implication, suppose that AD — DA = 0, Let e denote the fth standard basis vector of C . Then 0 = AD — DAid = DuAci — DAa. So Aci is an eigenvector of D with eigenvalue D. Because Da Djj unless i = j, it follows that Aa must be a scalar multiple of a for each L Hence A must be diagonal. ... 

Exercise 2.1 Consider the set of homogeneous polynomials in two variables with real coefficients. There is a natural addition of polynomials and a natural scalar multiplication of a polynomial by a complex number. Show that the set of homogeneous polynomials with these two operations is not a complex vector space. 

Exercise 2.3 Show that C (with the usual addition and multiplication) is itself a complex vector space of dimension 1. Then show thatC with the usual addition but with scalar multiplication by real numbers only is a real vector space of dim ension 2. 

Exercise 2.6 Let V be an arbitraty complex vector space of dimension n. Show that by restricting scalar multiplication to the reals one obtains a real vector space of dimension 2n. 

The functions in A form a complex vector space under the usual addition and scalar multiplication of functions. 

This fact will be at the heart of the proof of our main result in Section 6.5. Proof. First, we show that V satisfies the hypotheses of the Stone-Weierstrass theorem. We know that V is a complex vector space under the usual addition and scalar multiplication of functions adding two polynomials or multiplying a polynomial by a constant yields a polynomial. The product of two polynomials is a polynomial. To see that V is closed under complex conjugation, note that for any x e [—1, 1] and any constant complex numbers flo, , a sN[Pg.102]

Proposition 6.6 Suppose V is a finite-dimensional complex vector space with a complex scalar product. Suppose G, V, p) is a unitary representation. Suppose that every linear operator 7 V V that commutes with p is a scalar multiple of the identity. Then G, V, p is irreducible. 

Assume that a stress tensor T is known at a point. Assume further that there are three orthogonal surfaces passing through the point for which the stress vectors r are parallel to the outward-normal unit vectors n that describe the orientation of the surfaces. In other words, on each of these surfaces the normal stress vector is a scalar multiple of the outward-normal unit vector,... 

Square matrices and tensors can be characterized by their eigenvalues and eigenvectors. If M is an n x n square matrix (or tensor), there is a set of n special vectors, e, each with its own special scalar multiplier A for which matrix multiplication of a vector is equivalent to scalar multiplication of a vector ... 

Multiplication of the Dirac characters produces a linear combination of Dirac characters (see eq. (4.2.8)), as do the operations of addition and scalar multiplication. The Dirac characters therefore satisfy the requirements of a linear associative algebra in which the elements are linear combinations of Dirac characters. Since the classes are disjoint sets, the Nc Dirac characters in a group G are linearly independent, but any set of N< I 1 vectors made up of sums of group elements is necessarily linearly dependent. We need, therefore, only a satisfactory definition of the inner product for the class algebra to form a vector space. The inner product of two Dirac characters i lj is defined as the coefficient of the identity C in the expansion of the product il[ ilj in eq. (A2.2.8),... 

Identifying the operation Div as a scalar multiplication of the operator J by the flux vector, one rewrites the continuity equation (4.17) as... 

For example, a matrix of rank 1 can be formulated as the outer product of two vectors, such as uvT. (The rank of this matrix is 1 because all rows are scalar multiples of one other). Applying condition [46] to this update form B +1 = Bk + uvT, we obtain the condition that u is a vector in the direction of (yk - Bksk). If yk = Bksk,Bk already satisfies the QN condition [46]. Otherwise, we can write the general rank 1 update formula as ... 

Let us consider an operation that transforms the unit vector tij in other Hi. The vector determined in this way is a scalar multiple of nf. 

A linear vector. space is a set L containing elements (vectors) which can be related by two operations, addition and scalar multiplication, satisfying the conditions... 

Definition 38 A linear subspace of L is a subset of L that forms a linear vector space under the rules of addition and scalar multiplication defined for L. 

Multiplication of two matrices can be either scalar multiplication or vector multiplication. Scalar multiplication of two matrices consists of multiplying corresponding elements, i.e.. 

Vector Addition, Subtraction and Scalar Multiplication using Algebra... 

Because the product (3.8) is a probability, it must integrate to 1.0 when all possible outcomes are taken into account. Consequently, the wave functions are multiplied by arbitrary constants (n ),/2 chosen to make this integral come out to 1.0 over the complete range of motion. These are called normalization constants. It is legitimate to multiply solutions to the Schroedinger equation by an arbitrary constant because they are elements of a closed binary vector space. Multiplication of a solution by any scalar yields another element in the space, hence the product of the normalization constant and the wave function (or any other state vector in Hilbert space) also is a solution. 

Vectors The collection of all vectors containing n elements satisfying basic axioms (see, for example, Ramkrishna and Amundson (1985)) is called a linear space denoted by 91 . The basic axioms define scalar multiplication, vector addition (from which evolves the concept of linear combination), and a null vector that has all elements zero. Thus a linear combination of vectors Xj, j = is expressed as the vector E ia Xj, where the a/s are numbers. If the a/s are all... 

Q will be orthogonal. Here, v represents a normalized vector, T is an upper triangular matrix with Tjj = i - m and off-diagonal elements equal to 1. The symbol indicates a form of element by element scalar multiplication across... 

These vectors, thus, constitute an orthonormal set with respect to scalar multiplication. To express the scalar product A B in terms of the components of A and B, write... 

The set of elements, each made of n components in an order for which vector addition and scalar multiplication are defined. For example, an element x is given by... 

If the value of the effluent concentration C is known, the associated rate vector r(C) may be evaluated. From liquation 4.9, it is clear that r(C) and v are scalar multiples of each other by t. Moreover, since t can only assume positive values, r(C) and v must both point in the same direction. Hence for any point C satisfying Equation 4.9, this point must exist as a CSTR solution for the feed point Cf. We therefore arrive at the following result for CSTRs Fora specified feed point and rate expression r( C), the... 

A subspace of a vector space is a nonempty subset that is also a vector space. That is, vector subspaces also obey the laws of vector addition and scalar multiplication. If x and y are two vectors that lie in a vector subspace, then linear combinations of x and y will produce vectors that also lie in the subspace. For example, linear combinations of vectors [0, 0, 1] and [2, 0, 0] produce vectors that lie in a two-dimensional subspace (a plane) in R linear combinations of vector [5, 0.2, -3, 1, 8] lie in a one-dimensional subspace (a line) in R. ... 

Geometrically, two vectors x and y are linearly independent when X and y do not lie on the same line (they are not scalar multiples of each other). Linearly independent vectors may be used to generate (span) a vector space. For example, vectors aj, E2 e will span a two-dimensional subspace in if aj and a2 are linearly independent, and... 

Multiplication of one screw tiansfcMmation with a general spatial vector requires (10 scalar multiplications, 6 scalar additions). Thus, the complete transformation of a genoal vector requires a total of (20 multiplications, 12 additions) if two screw transformations are used. The product of a genoal transformation between adjacent coordinate systems and a general vector, on the otho- hand, requires (24 multiplications, 18 additions). 

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