Consider a vector space $V$ , show that: "$dimV=1\Leftrightarrow V$ has exactly two subspaces".
Well I can understand that since $dimV=1$ the only subspaces are the zero and it self, but how can i prove that? How can I prove the equivalence? Thanks in advance.
[Math] One-dimensional vector space and it’s subspaces
vector-spaces
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Best Answer
Suppose that dim$V = 1$. Let $V = span\{x\}$. If $U \leq V$ is a subspace, either $U$ is zero, or has a multiple of $x$, in which case by linearity it is $V$. So $V$ can only have $2$ subspaces indeed.
On the other hand, assume that the linear space $V$ has only two subspaces; then naturally these are $V$ and the zero subspace. If dim$V > 1$ then let $\{x_n\}$ be a linearly independent set of more than 1 vector. Selecting two of these and taking the span of each yields two non-zero subspaces, which must then both be $V$. Hence these aren't linearly independent, and so dim$V = 1$.