Lecture 13 Phys 3750 D M Riffe -1- 2/28/2013 Vector Spaces / Real Space Overview and Motivation: We review the properties of a vector space. As we shall see in the next lecture, the mathematics of normal modes and Fourier series is intimately related to the mathematics of a vector space. Key Mathematics: The concept and properties of a vector space, including addition, scalar multiplication, linear independence and basis, inner product, and orthogonality. I. Basic Properties of a Vector Space You are already familiar with several different vector spaces. For example, the set of all real numbers forms a vector space, as does the set of all complex numbers. The set of all position vectors (defined from some origin) is also a vector space. You may not be familiar with the concept of functions as vectors in a vector space. We will talk about that in the next lecture. Here we review the concept of a vector space and discuss the properties of a vector space that make it useful. A. Vector Addition. A vector space is a set (of some kind of quantity) that has the operation of addition (+) defined on it, whereby two elements v and u of the set can be added to give another element w of the set, 1 vuw += . (1) There is also an additive identity included in the set; this additive identity in known as the zero vector 0, such that for any vector v in the space v0v =+ . (2) The addition rule has both commutative uvvu +=+ (3) and associative () ( )wvuwvu++=++ (4) properties. 1 We denote vector quantities by boldface type and scalars in standard italic type. This is standard practice in most physics journals.Lecture 13 Phys 3750 D M Riffe -2- 2/28/2013 B. Scalar Multiplication The vector spaces that we are interested in also have another operation defined on them known as scalar multiplication, in which a vector u in the space can be multiplied by either a real or complex number a , producing another vector in the space uv a=. If we are interested in multiplying the elements of the space by only real numbers it is known as a real vector space; if we wish to multiply the elements of the space by complex numbers, then the space is known as a complex vector space. Scalar multiplication must satisfy the following properties for scalars a and b and vectors u and v , ()uuu baba +=+ , (5a) ()()uu abba = , (5b) ()vuvu aaa +=+ , (5c) uu =1 , (5d) 0u =0 . (5e) None of these properties should be much of a surprise (I hope!) C. Linear Independence and Basis The span of a subset of m vectors is the set of all vectors that can be written as a linear combination of the m vectors, mmaaa uuu +++ K2211. (6) The subset of m vectors is linearly independent if none of the subset can be written as a linear combination of the other members of the subset. If the subset is linearly dependent then we can write at least one of the members as a linear combination of the others, for example 112211 −−+++=mmmaaa uuuu K . (7)Lecture 13 Phys 3750 D M Riffe -3- 2/28/2013 For a given vector space if there is a maximum number of linearly independent vectors possible, then that number defines the dimension N of the vector space. 2 That means if we have identified N linearly independent vectors then those N vectors span the entire vector space. This means that any vector in the space can be written as a linear combination of the set of N independent vectors as NNvvv uuuv +++= K2211, (8) or in more compact notation as ∑==Nnnnv1uv . (9) Furthermore the coefficients3 nv in Eq. (9) are unique. The vectors nu in Eq. (9) are said to form a basis for the space. Now Eq. (9) should look strangely familiar. We have some quantity on the lhs that is written as a linear combination of quantities on the rhs. Hum… And you might even ask, assuming that I know the vectors nu in Eq. (7), how do I find the coefficients nv ? D. Inner Product This last question is most easily answered after we define one more operation on the vector space, known as the inner product of two vectors, which we denote ()vu, . The inner product returns a scalar, which is a real number for a real vector space or a complex number for a complex vector space. The inner product can be defined in any manner as long as it satisfies the following relationships ()()∗= uvvu ,, (10a) ()()()uwvwuvw ,,, baba +=+ (10b) Note the complex-conjugate symbol in Eq. (10a). If we are dealing with a real vector space, then we can just ignore the complex-conjugate symbol. Also note that Eq. (10a) implies that the inner product of a vector u with itself is a real number. It can be shown that Eqs. (10a) and (10b) imply that 2 If there is not a maximum number of linearly independent vectors, then the space is said to have infinite dimension. 3 The coefficients nv are also known as the scalar components of v in the basis {}Nuu ,,1K .Lecture 13 Phys 3750 D M Riffe -4- 2/28/2013 ( ) ()()wuwvwuv ,,,**baba +=+ (10c) ()()uvuv ,ba,ba*= (10d) In physics we are usually interested in vector spaces where ()0≥uu, , ()0=uu, iff 0u = . (10e) Such vector spaces are said to have a positive semi-definite norm (the norm is defined below). With these properties of the inner product denoted, we can define the concept of orthogonality. Two nonzero vectors u and v are said to be orthogonal if their inner product vanishes, i.e., if ()0=vu,. Note that if two vectors are orthogonal, then they are linearly independent. This is easy to see, as follows. Assume the converse, that they are linearly dependent. Then their (assumed) linear dependence means that vu a=, where a is some scalar [see Eq. (7)]. Then the scalar product ()()()vvvvuv ,,, aa== cannot be zero because v is not zero [see Eq. (10e)]. Thus they must be linearly independent. The converse is not true, two linearly independent vectors need not be orthogonal. The proof is given as one of the exercises. One last thing regarding the inner product. The quantity ()uuu ,= is generally known as the norm (or size) of the vector u. Often we are interested in vectors whose norm is 1. We can "normalize" any vector u with scalar multiplication by calculating ()uuuu,=ˆ. (11) The "hat" over a vector indicates that the vector's norm is 1. E. Orthogonal Basis Most of the time that we deal with a basis, the vectors in that basis are orthogonal. That is, their inner products with each other