For example,
let S = {$(x_1, x_1)| x_1 \in \mathbb R$} be a subspace of $\mathbb R^2$.
By definition, dim(S) = 1, and dim($\mathbb R^2$) = 2.
Then the set {(1, 1)} only has one vector, so is it linearly independent in S, but is linearly dependent in $\mathbb R^2$?
I know this doesn't make any sense, but we learned in class that a set of vectors can only be linearly independent if it spans the vector space that it is in.
Since (1,1) is in both S and $\mathbb R^2$, but the set {(1, 1)} only spans S, how come it is not only linearly independent in S and linearly dependent in $\mathbb R^2$ (since {(1,1)} does not span $\mathbb R^2$).
Sorry for this stupid question
Best Answer
You said you “learned in class that a set of vectors can only be linearly independent if it spans the vector space that it is in.” This isn’t correct, unless “the vector space that it is in” means “the smallest vector space that it is in,” which would not be a typical reader’s understanding. But even with that understanding, the statement is not useful, because the fact is not special to linearly independent sets. Every set of vectors spans the smallest vector space that contains them: a set of vectors spans its span. (That’s practically the definition of span.)
I suspect what you were supposed to learn in class was that “A set of vectors in a vector space $V$ can only be a basis for $V$ if the set spans $V$.” (In addition, it must be a linearly independent set of vectors.)
In particular, the set $\{(1,1)\}$ is a linearly independent set, whether it is considered as a set of vectors in $\mathbb R^2$ or as a set of vectors in what you call $S$. The fact that $\{(1,1)\}$ does not span $\mathbb R^2$ does not tell you anything about the linear independence of the set. (And by the way, any set containing only one vector is a linearly independent set of vectors so long as that one vector is not zero.)