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Dear friends I wanted to ask if $\operatorname{span}\{u_1,u_2,\dots,u_n\}$ is contained in
$\operatorname{span}\{v_1,v_2,\dots,v_n\}$ and both $u_i$'s and $v_i$'s are linearly independent vectors in the vector space $V$, can we say both spans are equal?

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The answer is yes. Span($u_1,\ldots, u_n$) is a subspace $A$ of $V$, and Span($v_1,\ldots,v_n$) is a subspace $B$ of $V$. Your conditions imply that $A\subseteq B$, and that $dim(A)=n=dim(B)$. Hence they are equal.

A quick proof of this: If $A\neq B$, then there is some $x\in B\setminus A$. We must have $x\notin Span(u_1,\ldots, u_n)$. But then $C=Span(u_1,\ldots,u_n,x)$ would have higher dimension than $A$, and still $C\subseteq B$.

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