Using the given basis of a vector to prove other basis

[Kronos1]

Hi Guys having a bit of trouble understanding vector basis.

If $$\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$$ is a basis for vector space $V$ over the field $F$

and $${f}_{1}=-{e}_{1}, {f}_{2}={e}_{1}-{e}_{2}, {f}_{3}={e}_{1}-{e}_{3}$$

how can I go about proving that $$\left\{{f}_{1},{f}_{2},{f}_{3}\right\}$$ is also a basis for $V$?

 

[Euge]

Hi Guys having a bit of trouble understanding vector basis.

If $$\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$$ is a basis for vector space $V$ over the field $F$

and $${f}_{1}=-{e}_{1}, {f}_{2}={e}_{1}-{e}_{2}, {f}_{3}={e}_{1}-{e}_{3}$$

how can I go about proving that $$\left\{{f}_{1},{f}_{2},{f}_{3}\right\}$$ is also a basis for $V$?

Hi Kronos,

Solve the system of equations for $e_1$, $e_2$, and $e_3$. Your resulting equations will show that $e_1$, $e_2$, and $e_3$ all belong to the linear span of $f_1$, $f_2$, and $f_3$. Thus Span($e_1$, $e_2$, $e_3$) $\subset$ Span($f_1$, $f_2$, $f_3$). Since $e_1$, $e_2$, and $e_3$ form a basis for $V$, they span $V$ and thus $V = \text{Span}(f_1, f_2, f_3)$. This means that $f_1$, $f_2$, and $f_3$ span $V$. The $f$-vectors also linearly independent since $V$ is three-dimensional (as $e_1$, $e_2$, and $e_3$ form a basis for $V$).