Multiplication by a Scalar

The related operation of division by the scalar A will be denoted by [yl]/A. The elements of this matrix are A j/K. This operation is equivalent to multiplication of [yl] by 1/A. 

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Definition.—The set of all vectors derivable from any one vector in by multiplication by a scalar is called a ray in 3P. 

In Subheading 2.3. the important class of vectors with continuous-valued components is described. A number of issues arise in this case. Importantly, since the objects of concern here are vectors, the mathematical operations employed are those applied to vectors such as addition, multiplication by a scalar, and formation of inner products. While distances between vectors are used in similarity studies, inner products are the most prevalent type of terms found in MSA. Such similarities, usually associated with the names Carbo and Hodgkin, are computed as ratios, where the inner product term in the numerator is normalized by a term in the denominator that is some form of mean (e.g., geometric or arithmetic) of the norms of the two vectors. 

Finally, we must show that k is unique and T is unique up to multiplication by a scalar of modulus one. Suppose Ti, /ci and 72, a 2 both satisfy the requirements of the proposition. We must show that /ci = k 2 and there is a real number d such that Ti = e" T2. know that for any element v e we have [T lKiCv)] = [T2/e2(r )]- Applying the physical symmetry to both sides we find that... 

The definition of a mathematical space begins with the set of objects X, Y, Z,. .. that occupy the space (an intrinsically empty space being a physically problematic concept). Among the simplest algebraic structures that can characterize such objects is that of a linear manifold, also called a linear vector space, affine space, etc. By definition, such a manifold has only two operations— addition (X + Y) and multiplication by a scalar (AX)— resulting in each case in another element of the manifold. These operations have the usual distributive,... 

Multiplication of a matrix A by a scalar x follows the rules one would expeet from the algebra of numbers Eaeh element of A is multiplied by the sealar. If... 

Postulate A.—3P is linear. By this is meant (i) the vectors of JP are such that we can define the sum of any two of them, the result being also a vector in o> + 6> = c> (ii) they are such that a meaning can be ascribed to multiplication of any vector in by a scalar complex number, the result being also a vector in. In particular,... 

These relationships that express intensive parameters in terms of independent extensive parameters, are called equations of state. Because of the homogeneous first-order relationship between the extensive parameters it follows that multiplication of each of the independent extensive parameters by a scalar A, does not affect the equation of state, e.g... 

If a matrix is to be multiplied by a scalar x, the multiplication is performed on every element of the matrix. Obviously, this operation is commutative. 

The simplest nontrivial example of a complex vector space is C itself. Adding two complex numbers yields a complex number multiplication of a vector by a scalar in this case is just complex multiplication, which yields a complex number (i.e., a vector in C). Mathematicians sometimes call this complex vector space the complex line. One may also consider C as a real vector space and call it the complex plane. See Figure 2.1. 

A matrix may be multiplied by a scalar number or by another matrix. For multiplication of a matrix [c,y] by a scalar, ct, we have... 

Scalar multiplication of a matrix A by a scalar nmnber A is carried out by multiplying all the elements by A ... 

Multiplication of a complex number by a scalar (real number) is achieved by simply multiplying the real and imaginary parts of the complex number by the scalar quantity. Multiplication of two complex numbers is performed by expanding the expression (a + ih) c + id) as a. sum of terms, and then collecting the real and imaginary parts to yield a new complex number. 

Similarly, if P = cS, where c is a constant, then we find (Prob. 7.52) Pjf. = cSjk, which is the rule for multiplication of a matrix by a scalar. 

Thus, addition (or subtraction) of two matrices is accomplished by adding (or subtracting) the corresponding elements of the two matrices. These operations may, of course, be extended to any number of matrices of the same order. In particular, observe that adding k identical matrices A (equivalent to scalar multiplication of A by k) results in a matrix in which each element is merely the corresponding element of A multiplied by the scalar k. Thus, when a matrix is to be multiplied by a scalar, the required operation is done by multiplying each element of the matrix by the scalar. For example, for A given in relation (A.2),... 

Strictly speaking, these vectors should be called geometric vectors since they do not, in all cases, satisfy the properties of algebraic vectors (e.g., algebraic vectors satisfy the axioms of a linear vector space, namely, the addition of two vectors or the multiplication of a vector by a scalar should result in another vector that also lies in the space). Nevertheless, the terminology vector, which is common in chemical informatics, will be used here to include both classes of vectors. 

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