Relation of basis of vector space and basis of its quotient space

Question

Let $V$ be a vector space over a field $\mathbb{F}$ of finite dimension $n$. Let $W$ be a subspace with a basis $B_1=\{\mathbf{w}_1,\ldots,\mathbf{w}_r\}$. Then we have:

$B_1\coprod\{\mathbf{u}_{r+1},\ldots,\mathbf{u}_n\}$ is a basis of $V$ if and only if the cosets $\{\overline{\mathbf{u}}_{r+1},\ldots,\overline{\mathbf{u}}_n\}$ is a basis of quotient space $V/W$.

I have successfully shown the forward direction. However, I have little idea how to proceed to prove the backward direction. Could any kind soul enlighten me?

Answer 1

Let $v \in V$. Then $\overline{v}$ can be written uniquely as a linear combination of $\overline{\mathbf{u}}_{r+1},\ldots,\overline{\mathbf{u}}_n$, say $$ \overline{v} = a_{r+1} \overline{\mathbf{u}}_{r+1}+ \dots + a_{n} \overline{\mathbf{u}}_n. $$ Thus $$ \overline{v - a_{r+1} {\mathbf{u}}_{r+1}+ \dots - a_{n} {\mathbf{u}}_n} = \overline{0}, $$ that is $$ v - a_{r+1} {\mathbf{u}}_{r+1}+ \dots - a_{n} {\mathbf{u}}_n \in W. $$ As such, this can be written uniquely as a linear combination of $\mathbf{w}_1,\ldots,\mathbf{w}_r$, so that $$ v = a_{1} \mathbf{w}_1 + \dots + a_{r} \mathbf{w}_r +a_{r+1}{\mathbf{u}}_{r+1}+ \dots + a_{n} {\mathbf{u}}_n $$ uniquely.