Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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2. Check that the following are not vector spaces by proving that at least one axiom in the definition of a vector space is false. (a) The quadruple (R, R", H, O2). Here, H denotes the usual operation of addition multiplication among vectors, while O2 denotes the nonstandard operation of scalar multiplication given by c(x1, x2, . .. , Xn) = (cx1,0, 0, . . .,0). (b) The quadruple (R, V, H, D), where V denotes in this case the subset of Rmxn whose elements have the entry in row 1 and column 1 equal to some r E R, r + 0. Here, B and O denote the usual operations of addition and scalar multiplication among matrices.
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Transcribed Image Text:2. Check that the following are not vector spaces by proving that at least one axiom in the definition of a vector space is false. (a) The quadruple (R, R", H, O2). Here, H denotes the usual operation of addition multiplication among vectors, while O2 denotes the nonstandard operation of scalar multiplication given by c(x1, x2, . .. , Xn) = (cx1,0, 0, . . .,0). (b) The quadruple (R, V, H, D), where V denotes in this case the subset of Rmxn whose elements have the entry in row 1 and column 1 equal to some r E R, r + 0. Here, B and O denote the usual operations of addition and scalar multiplication among matrices.
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