I have a simple multiple choice I've been trying to understand here for a while now.

Suppose that $F$ is spanned by the functions $f_1,f_2,f_3,f_4$. What can we conclude about $\text{dim}(V)$?

a) $\text{dim}(V)=4$

b) $\text{dim}(V)\geq 4$, but we know nothing else

c) $\text{dim}(V)\leq 4$, but we know nothing else

d) We can conclude nothing about $\text{dim}(V)$

e) None of the above

NORMALLY I would say that the answer is $b)$ but that only holds for a finite dimensional vector space. I'm not sure if I can assume that $b$ is true if $F$ is infinite dimensional.

Can someone guide me on this? Thanks!

  • $\begingroup$ The answer is c). There is no infinite dimensional space in this question. $\endgroup$ – Kavi Rama Murthy Oct 19 '20 at 5:56
  • $\begingroup$ Okay so I need to ask. Why? I thought since F could be anything, it's infinite dimensional. Also why is it c? For that to happen, don't the vectors have to all be linearly independent? $\endgroup$ – Future Math person Oct 19 '20 at 5:58

If $f_1,f_2,f_3,f_4$ are linearly independent, then $\{f_1,f_2,f_3,f_4\}$ is a basis of $F$, hence $\dim (V)=4.$

If $f_1,f_2,f_3,f_4$ are linearly dependent, then $ \dim(V) \le 3.$

The answer is c).

  • $\begingroup$ How do you know the vectors are linearly dependent? Is it because they span $F$ so it's possible to express one or more vectors as a linear combination of each other and hence the dimension can be at most $3$? $\endgroup$ – Future Math person Oct 19 '20 at 6:04

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