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  "body": "\n## Proof\n\nWe extend a basis of $W$ to a basis of $V$ and show that the cosets of the new vectors form a basis of $V / W$.\n\n1. Let $n = \\dim V$. By the [[dimension of a subspace|0QW084460E]], $W$ is finite-dimensional. Let $m = \\dim W$ and let $(w_1, \\dots, w_m)$ be a [[basis|0KCQ9Y3QKA]] of $W$.\n2. The family $(w_1, \\dots, w_m)$ is [[linearly independent|0CRMGPAH83]] in $V$. By [[extension to a basis|0M4SXE5WW3]], it is part of a basis of $V$.\n3. By the [[dimension theorem|0WK0Y8RH1M]], this basis has $n$ vectors. Write it as $(w_1, \\dots, w_m, v_{m+1}, \\dots, v_n)$.\n4. In the [[quotient space|0BR3FY5Q8M]] $V / W$, the zero vector is $0 + W = W$, since $(0 + W) + (v + W) = v + W$ for all $v \\in V$.\n5. For $x \\in V$, we have $x + W = W$ if and only if $x \\in W$. If $x + W = W$, then $x = x + 0 \\in W$. If $x \\in W$, then $x + w \\in W$ and $w = x + (w - x)$ for each $w \\in W$, so $x + W = W$.\n6. The cosets $v_{m+1} + W, \\dots, v_n + W$ [[span|088RYKAWK9]] $V / W$. Let $v + W \\in V / W$, and write $v = \\sum_{i=1}^m a_i w_i + \\sum_{k=m+1}^n a_k v_k$ in the basis of step 3.\n7. By the operations of $V / W$, $v + W = \\sum_{i=1}^m a_i (w_i + W) + \\sum_{k=m+1}^n a_k (v_k + W)$.\n8. Each $w_i$ lies in $W$, so $w_i + W$ is the zero vector by steps 4 and 5. Hence $v + W = \\sum_{k=m+1}^n a_k (v_k + W)$.\n9. The cosets $v_{m+1} + W, \\dots, v_n + W$ are linearly independent. Suppose $\\sum_{k=m+1}^n b_k (v_k + W) = W$ with $b_k \\in F$.\n10. The left side equals $x + W$ with $x = \\sum_{k=m+1}^n b_k v_k$. By step 5, $x \\in W$.\n11. Since $(w_1, \\dots, w_m)$ spans $W$, $x = \\sum_{i=1}^m d_i w_i$ for some $d_1, \\dots, d_m \\in F$.\n12. Then $\\sum_{i=1}^m d_i w_i - \\sum_{k=m+1}^n b_k v_k = 0$. The basis of step 3 is linearly independent, so all $b_k = 0$.\n13. By steps 6 to 12, $(v_{m+1} + W, \\dots, v_n + W)$ is a basis of $V / W$ with $n - m$ vectors.\n14. So $V / W$ is [[finite-dimensional|0ZHWKQMVX5]] with [[dimension|03GSZ8RTZJ]] $n - m$. Hence $\\dim(V / W) + \\dim W = (n - m) + m = \\dim V$.\n\n## Idea\n\nChoose a basis of $V$ that starts with a basis of $W$. Passing to the quotient kills the vectors of $W$. The remaining basis vectors survive as cosets and form a basis of $V / W$, so exactly $\\dim W$ dimensions are lost.\n",
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