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Be $X$ a vectorial space of size $n$, $n \ge 1$ and $u_1,u_2,\ ... \ u_n$ vectors from $X$.

Prove that these affirmations are equivalent:

  1. $(u_1,u_2...u_n)$ is linear independent;
  2. $(u_1,u_2...u_n)$ spans $X$;
  3. $\{u_1,u_2...u_n\}$ is a basis of $X$.

Please help me! I don't know how to start. Please give me a hint or help. Thank you very much!

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Of course, 3. $\implies$ 1. and 3. $\implies$ 2. by the definition of basis.

If 1. holds, then $\langle u_1,\ldots,u_n\rangle$ has dimension $n$. But $\dim X=n$. Therefore $\langle u_1,\ldots,u_n\rangle=X$ and so $\{u_1,\ldots,u_N\}$ is a basis.

If 2. holds, suppose that $\{u_1,\ldots,u_n\}$ is linearly dependent. But then one of the vectors is a linear combination of all others, and so you can forget it. So, $\dim\langle u_1,\ldots,u_n\rangle<n$, which is impossible. Therefore, $\{u_1,\ldots,u_n\}$ is linearly dependent and so it's a basis.

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They are equivalent from the definition and properties of a basis in a n-dimensional vector space, i.e. a set of n vectors linearly independent which span the space.

For details you can look in any book about the subject.

http://mathworld.wolfram.com/VectorBasis.html

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