Let V be the vector space over the field of real numbers consisting of all real-valued sequences (a1, a2, az, ...). Indicate, without proof, which of the following subsets of V is a vector space. (a) The set of all sequences satisfying ai = a2 = a3. (b) The set of all sequences that are increasing, i.e., for which ai s a2 < a3 < ….. (c) The set of unbounded sequences.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Q1
Let V be the vector space over the field of real numbers consisting of all
real-valued sequences (a1, a2, a3, ...). Indicate, without proof, which of
the following subsets of V is a vector space.
(a) The set of all sequences satisfying a1 = a2 = a3.
(b) The set of all sequences that are increasing, i.e., for which
a1 < a2 < az < …
(c) The set of unbounded sequences.
(d) The set of bounded sequences.
(e) The set of sequences satisfying an+1 = an + an-1 for all n > 2.
(f) The set of sequences satisfying an+1 = an × an-1 for all n > 2.
Some comments:
(1) You do not need to justify your answer, just write which of (a)-(f) are
vector spaces and which are not.
(ii) A sequence (a,)nz1 is said to be bounded if there is a real number
B > 0 (depending on the sequence) for which all a, lie in the interval
[-B, B]. It is said to be unbounded otherwise.
Transcribed Image Text:Q1 Let V be the vector space over the field of real numbers consisting of all real-valued sequences (a1, a2, a3, ...). Indicate, without proof, which of the following subsets of V is a vector space. (a) The set of all sequences satisfying a1 = a2 = a3. (b) The set of all sequences that are increasing, i.e., for which a1 < a2 < az < … (c) The set of unbounded sequences. (d) The set of bounded sequences. (e) The set of sequences satisfying an+1 = an + an-1 for all n > 2. (f) The set of sequences satisfying an+1 = an × an-1 for all n > 2. Some comments: (1) You do not need to justify your answer, just write which of (a)-(f) are vector spaces and which are not. (ii) A sequence (a,)nz1 is said to be bounded if there is a real number B > 0 (depending on the sequence) for which all a, lie in the interval [-B, B]. It is said to be unbounded otherwise.
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