Two basis for a vector Space $V$ has the same coordinates. Does that follow both basis are identical?

Question

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?

Answer 1

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.

Answer 2

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 $\beta_2$.