Determine whether each set equipped with the given operations is avector space. For those that are not vector spaces identify the vector space axioms that fail. a). The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard operations on R^2. b). The set of all triples of real numbers with the standard vectoraddition but with scalar multiplication deﬁned by k (x, y, z) = (k^2x, k^2y, k^2z) c). The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication

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Mathematics

Camden Stein

3 October, 07:56

Determine whether each set equipped with the given operations is avector space. For those that are not vector spaces identify the vector space axioms that fail. a). The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard operations on R^2. b). The set of all triples of real numbers with the standard vectoraddition but with scalar multiplication deﬁned by k (x, y, z) = (k^2x, k^2y, k^2z) c). The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication

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Autumn Gray · Answer 1

Answer: Determine whether each set equipped with the given operations is avector space. For those that are not vector spaces identify the vector space axioms that fail. a). The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard operations on R^2. b). The set of all triples of real numbers with the standard vectoraddition but with scalar multiplication deﬁned by k (x, y, z) = (k^2x, k^2y, k^2z) c). The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication