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03252010linalg

# 03252010linalg - Mathematics of Linear Algebra 28th January...

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Mathematics of Linear Algebra 28th January 2010 Elements of linear algebra play a dominant role in chemical applications. For the purposes of undergraduate physical chemistry courses, quantum me- chanics and select areas of thermodynamics can be formulated in terms of the elements of linear algebra. Thus, we present here a brief review of concepts of linear algebra. we first begin with vectors in Cartesian space (which everyone can conceptualize easily), and then we will generalize to generic vector spaces in anticipation of the use of linear algebra in quan- tum mechanics (think about how we would consider a function in quantum mechanics in terms of vectors?). 1 Linear Algebra in Cartesian Space 1.1 Vectors and Operations Vectors in Cartesian space are treated as: ~a = ~ e 1 a 1 + ~e 2 a 2 + ~ e 3 a 3 = i ~e i a i The vectors ~e i are termed a basis , and they represent a complete set of elements that can be used to describe all vectors. We are normally used to seeing them as the coordinate -x, -y, -z axes, but they can be any general mutually perpendicular unit vectors. Then, any vector in Cartesian space (3-space) can be written as a linear combination of the general basis vectors, i , i = 1 , 2 , 3. ~a = ~ 1 a 0 1 + ~ 2 a 0 2 + ~ 3 a 0 3 = i ~ i a 0 i Note that the coefficients have special meaning, and this will be reserved for later discussion. 1

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A vector is thus represented by the three components with respect to a basis . Using the 2 general basis presented above, we can write a vector ~a as a column matrix as: ~a = a = a 1 a 2 a 3 ~a 0 = a 0 = a 0 1 a 0 2 a 0 3 The scalar or dot product of two vectors ~a and ~ b is defined as ~a · ~ b = a 1 b 1 + a 2 b 2 + a 3 b 3 = X i a i b i Note that ~a · ~a = a 2 1 + a 2 2 + a 2 3 = | ~a | 2 Now let’s use our general definition of vectors to define the scalar prod- uct. ~a · ~ b = X i X j ~ e i · ~ e j a i b j Let’s expand this out. What do we obtain? In order for the full sum to equate to the operational definition of the dot product, we arrive at a condition we have to require of the basis vectors. This is namely: ~ e i ~e j = δ ij = δ ji = 1 if i = j 0 otherwise This is one way to state that the basis vectors are mutually perpen- dicular (orthogonal) and have unit length (normal) . They are or- thonormal . What is the projection of a vector ~a along one of the basis vectors ~e j ? The scalar product (dot product) is the operational equivalent of projecting a vector onto another vector (in this case projecting ~a onto ~e j ) as: 2
~ e j · ~a = X i ~e j a i = X i δ ij a i = a j We have used the orthonormality condition in the last step. Employing the concept of the scalar product as a projection of a vector onto a vector, we can write the general form of a vector as: ~a = i ~e i ~ e i · ~a = 1 · ~a The notation, 1 = i ~ e i ~e i is the unit dyadic . A dyadic is an entity that when dotted into a vector, leads to another vector. Ordinarily, a dot product of two vectors leads to a scalar. This is the distinction, and an important one.

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