Some subsets are subspaces, and some subsets are not subspaces. However as far as the other conditions go, the zero vector is in the complex numbers and a real number under vector addition with another real number or multiplication by a real scalar remains in the real numbers and therefore these conditions are satisfied. I am unsure of the effect of the "given a subset" terminology as regards what is allowed. I would argue no since the complex numbers are not a subset of the real numbers. Just to add an example to help me locate my problem since I find it difficult to word.Īre the set of complex numbers in one dimension under vector addition and real scalar multiplication a subspace of the real numbers in one dimension. Definition 1 (Subspace) A subspace W of a vector space V is a subset of V that is closed under the addition and scalar multiplication operations on V. Hence I thought S was not an subspace of R 2. However as far as I can tell (5i,0) ε S but (5i,0) is not an element of R 2. I was in contact with my lecturer and he said that it is implied in the question ε R and these are the values of S which the later tests for closure and the zero value will test. Again the constraint allows for the use of any complex numbers as far as I can tell and hence is not a subspace since it is not a subset of r 3 In the context of our Hilbert space, if for every sequence ( x n) in M with a limit x H, we have x M, then M is closed. This one is closed under vector addition and scalar multiplication and contains the zero vector. An equivalent definition for a set M to be closed in a metric space is that if ( x n) is a sequence of elements in M that is convergent, then the limit is in M (i.e. I thought S was not a subset, since I can choose any complex numbers x 1 =(a+bi) x 2 = (c+di) such that the following values are true and they will be contained within the set S: X 1 = 5, x 2 =1, would be an element of S but Not closed under vector addition because: Question 1) S is not a subspace since it is not a subset and is also not closed under vector addition or scalar multiplication. Ok, I thought that neither of these were subspaces of R 3 so my reasoning is as follows: S is a subset of T which is also a vector space under the same operations. Question 2.) S is the set of all vectors such that:ĭefinition of subspace: A subspace S of a vector space T is a subset which contains the zero vector and is closed under the vector addition and scalar multiplication which define T. Closure under addition: If u and v are in V, then u. Question 1.) S is the set of all vectors such that x 1 2 + x 2 2 < 36 A subspace of R n is a subset V of R n satisfying: Non-emptiness: The zero vector is in V. The questions ask if a set S is a subset R n. The original printed problems can be found as attachments.
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