I want to know If n vectors is linearly independent, their dimension is n

Question

is it always possible? (I don't know the basic concept)

My question is same as "If n vectors are linearly independent, is their span R^n" this question??

Answer 1

Yes if the vectors have $n$ coordinates ! It's true ! By the definition of being independent. Find the solution here : If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?

Answer 2

If you have $n$ linearly independent vectors in $\mathbb R^n$ the span is exactly $\mathbb R^n$ whereas, if you have $n$ linearly independent vectors in some real vector space $V$ then the span of those vectors is linearly isomorphic to $\mathbb R^n$.

In general, in any vector space $V$ over a field $\mathbb F$ span of any $n$-linearly independent vectors form a subspace $\mathbf{isomorphic}$ to $\mathbb F^n$.