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MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9

Tham khảo tài liệu 'mathematical method in science and engineering episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | GROUP SPACES 273 5. For each F E V there exists an inverse T such that T Ũ 0. 6. Multiplication of a vector T with the number one leaves it unchanged IT If. 7. A vector multiplied with a scalar is another vector elf G V. A set of vectors If i G V i 1 2 . n is said to be linearly independent if the equality Cl- c2 2 4----Ecnlfn o 11.362 can only be satisfied for the trivial case ci C2 --- cn 0. 11.363 In an A-dimensional vector space we can find N linearly independent unit basis vectors êị V i 1 2 . n such that any vector in V can be expressed as a linear combination of these vectors as if Cjêi c2ê2 4----1- c en . 11.364 11.14.2 Inner Product Space Adding to the above properties a scalar or an inner product enriches the vector space concept significantly and makes physical applications easier. In Cartesian coordinates the inner product also called the dot product is defined as ifi ữ 2 ĩfỵ if 2 vlxv2x vlvv2v vlzv2z. 11.365 Generalization to arbitrary dimensions is obvious. The inner product makes it possible to define the norm or magnitude of a vector as ỊVỊ if I 1 2 11.366 where the angle between two vectors is defined as COSỚỊ2 X.V .X.2. 11.367 1 11 I Basic properties of the inner product are If 2 if 2 ifi 11.368 and 11 1 alf2 4- blf3 a 1ft if 2 b 1 lf3 11.369 where a and b are real numbers. A vector space with the definition of an inner product is also called an inner product space. 274 CONTINUOUS GROUPS AND REPRESENTATIONS 11.14.3 Four-Vector Space In Section 10.10 we have extended the vector concept to Minkowski spacetime as four-vectors where the elements of the Lorentz group act on four-vectors and transform them into other four-vectors. For four-vector spaces properties 1 7 still hold however the inner product of two four-vectors Aa and Ba is now defined as AaBa ga0AaB0 11.370 AữBữ - A1B1 - A2B2 - A3B3 where ga 3 is the Minkowski metric. 11.14.4 Complex Vector Space Allowing complex numbers we can also define complex vector spaces in the complex plane. For .