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

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?

• Thank you for editing my question, I really gotta learn how to do proper math notation on here lol – user763771 Mar 26 '20 at 18:25

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.
• I mean the coordinates of $u_1$ with respect to both bases. – José Carlos Santos Mar 26 '20 at 19:59
• How does that affect x∈V? $u1$ is already part of basis beta 2. – user763771 Mar 26 '20 at 20:00
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 $$\beta_2$$.