I was confused about zero dimension vector subspaces. Can you please answer the following questions with details/examples.
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What is the dimension and basis for vector space that is just composed of the zero vector.
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If the answer to the latter is an empty set, how can I construct a zero vector of say n rows from the empty set?
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Say there are three vector spaces
[0] [0, 0] [0 ,0, 0]. What are the basis for them. Isn't just saying empty set led to ambiguity about the zero vector subspace, it is a basis for?
Best Answer
(1) A vector space that is composed of just the zero vector is zero dimensional and its basis is the empty set.
(2) You can construct a zero vector because the empty sum is defined to be zero (this is somewhat of a cheat). The sum $\sum_{v_i\in\emptyset}a_iv_i$ is an empty sum, and it is defined to be the zero element of the vector space.
(3) The vector spaces $\{[0]\}$, $\{[0,0]\}$, and $\{[0,0,0]\}$ are all isomorphic, so there really isn't much ambiguity (they are all, in essence the same space). If you want to use the sum from before, the empty sum valuates to the zero element in the vector space in which you're working.