If this is true, then every vector space must always have at least one subspace, the one consisting of only the zero vector, correct?
Thanks!
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By the definition of a vector space, yes. – obataku May 05 '13 at 01:26
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2The existence of a zero vector is in fact part of the definition of what a vector space is: this makes your question quite weird. – Mariano Suárez-Álvarez May 05 '13 at 09:59
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Yes, and yes, you are correct.
The existence of a zero vector is in fact part of the definition of what a vector space is.
Every vector space, and hence, every subspace of a vector space, contains the zero vector (by definition), and every subspace therefore has at least one subspace:
The subspace containing only the zero vector vacuously satisfies all the properties required of a subspace. It is closed under vector addition (with itself), and it is closed under scalar multiplication: any scalar times the zero vector is the zero vector.
amWhy
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Correct me if I am wrong, but isn't it one of the requirements for a vector space to have a zero vector? – imranfat May 05 '13 at 01:18
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Yes, it is. That's my first yes. My second yes is that the zero vector constitutes a subspace of every vector space. – amWhy May 05 '13 at 01:19
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Every vector space also contains itself, and is its own subspace.
Except in one very special situation (which one?) it follows from this that almost every vector space has two subspaces.
Mariano Suárez-Álvarez
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@JohnathonSvenkat, you tell me :-) Which are the only vector space which have one one subspace? – Mariano Suárez-Álvarez May 05 '13 at 22:58
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