Proposition I.16-2

Question

Let $V$ be a vector space over a field $\mathbb{K}$, and let $W$ be a vector subspace of $V$. If $\dim{V} = n$ and $\dim{W} = m$, show that $\dim{V/W} = n-m$.

Toğrul Topal · Accepted Answer

The property of the cosets that we will use a lot is: $[w + v] = [kw + v]$ for any $k \in \mathbb{K}$.\par
      Let $\{w_1,\ldots,w_m\}$ be the basis for $W$. By the extension theorem we can add $n-m$ vectors to get an extended basis $\{w_1,\ldots,w_m,v_1,\ldots, v_{n-m}\}$ of $V$. Now consider the collection of cosets $\{ [w + v_1], \ldots, [w+v_{n-m}]\}$ for some arbitrary $w\in W$. We argue that this collection of cosets of size $n-m$ constitutes for a basis for $V/W$. Pick an arbitrary coset $C \in V/W$. Then $C = [u]$ for some $u\in W$ and we can write
      \begin{align*}
        u &= a_1w_1 + \ldots + a_mw_m + a_{m+1}v_1 + \ldots + a_{n}v_{n-m}\
          &= w' +  a_{m+1}v_1 + \ldots + a_{n}v_{n-m}\
          &\in [w' +  a_{m+1}v_1 + \ldots + a_{n}v_{n-m}] = [(w' +  a_{m+1}v_1) + \ldots + (w' + a_{n}v_{n-m})]\
          &= [w' + a_{m+1}v_1] + \ldots + [w' + a_nv_{n-m}]\
          &= a_{m+1}[w' + v_1] + \ldots + a_n[w' + v_{n-m}]\
      \end{align*}
      In other words, we have represented $C$ as a linear combination of cosets $[w' + v_1], \ldots, [w' + v_{n-m}]$. Since our choice of $C$ was arbitrary, $[w' + v_1], \ldots, [w' + v_{n-m}]$ spans $V/W$. Furthermore, the cosets are linearly independent, since the only way that
      $$a_{m+1}[w' + v_1] + \ldots + a_n[w' + v_{n-m}] = W + a_{m+1}v_1 + \ldots + a_{n}v_{n-m}= W + 0 $$
      is true, is when $a_{m+1},\ldots,a_n$ are all zero (this follows by linear independency of $v_1,\ldots,v_{n-m}$).

Proposition I.16-2

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Proposition I.16-2

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