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Extra info for Advanced Calculus
102) and Rules XI11 alone. This leads to the following definition: DEFINITION A Euclidean n-dimensional vector space is a collection of objects u, v, . called vectors, including a zero vector 0, for which the operations of addition, multiplication by scalars, and scalar product are defined and obey Rules I through XIII. a By virtue of our discussion there is, except for notation, really only one Euclidean n-dimensional vector space, namely, Vn . We have emphasized the axiomatic approach to vectors.
V,, is nonsingular. Therefore the equation Ac = v n + has ~ a unique solution for c; that is, By Rule (a), v l , . . , vn+]would then be linearly dependent, contrary to assumption. Therefore there cannot be n 1 linearly independent vectors in Vn. + Proof of (i). Let vl , . . , vk be a basis for V". Then vl , . . , vk are linearly independent. For if c l v l . . ckvk = 0, then + + since, by definition of a basis, 0 can be represented in only one way as a linear combination of v l , . . Hence by Rules (d) and (g) we must have k _( n.
We carry this out for the general case: and, in general, for i = I , . . , m , where , C.. -a;lbl, - + . . +aipbp, fori = 1 , . m. j = 1, . . , n . 57). We write C = A B and have thereby defined the product of the matrices A and B. 14 Product of two matrices . We observe that (aiI , . . , a;,) is the ith row vector of A and that col (bl, , . . b,) is the jth column vector of B . Hence to form the product A B = C = (c,,), we obtain each c,, by multiplying corresponding entries of the ith row of A and the jth column of B and adding.