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Let $\beta_{1} = \{v_1,v_2,\ldots,v_n\}$ and $\beta_{2} = \{u_1,u_2,\ldots,u_n\}$ be two bases for some vector space $V$. If the coordinates for every vector $x\in V$ are identical with respect to both bases, does it follow that $v_i = u_i$ for all $i=1,2,\ldots, n$?

Since both coordinates are identical to each other, does that not force $v_i = u_i$? Am I approaching this right?

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  • $\begingroup$ Thank you for editing my question, I really gotta learn how to do proper math notation on here lol $\endgroup$ – user763771 Mar 26 '20 at 18:25
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The answer is affirmative. The coordinates of $u_1$ with respect to $\beta_1$ are $1,0,0,\ldots,0$ and therefore its coordinates with respect to $\beta_2$ are also $1,0,0,\ldots,0$. But this means that $u_1=v_1$. And the same argument applies to the other vectors.

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  • $\begingroup$ Do you mean that the coordinates of x with respect to Beta 1 and 2? Sorry I want to make sure I fully understand this $\endgroup$ – user763771 Mar 26 '20 at 18:26
  • $\begingroup$ I mean the coordinates of $u_1$ with respect to both bases. $\endgroup$ – José Carlos Santos Mar 26 '20 at 19:59
  • $\begingroup$ How does that affect x∈V? $u1$ is already part of basis beta 2. $\endgroup$ – user763771 Mar 26 '20 at 20:00
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Indeed, since in basis $\beta_1$, each $v_i$ has coordinates $(0,\dots,0,\underset{\substack{\uparrow\\i\text{-th}\\ \text{coordinate}}}{1},0,\dots,0)$, and it has the same coordinate in basis $\beta_2$.

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  • $\begingroup$ Does each v-i have coordinates (0,.....0,1,0,....0) or does each x vector in vector space V have (0,.....0,1,0,....0)? $\endgroup$ – user763771 Mar 26 '20 at 18:31

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