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

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??

• Can you please phrase your question in a more understandable manner? As it is, this does not really make sense. I presume that you wanted to ask ‘If a vector space is of dimension n and I have a set of n vectors that are linearly independent, does that set form a basis for the vector space?’. If that is what you wanted to ask, edit your question accordingly. Commented May 7, 2020 at 10:25

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 $\mathbb{R}^n$?

• thanks, Then, the comment"If the vectors have n coordinates"is not in the question the question is false? Commented May 7, 2020 at 10:34
• Hum... Yeah but you will figure out quickly. Commented May 7, 2020 at 15:29

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 $$\mathbb F^n$$.