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  "body": "\n## Proof\n\n1. Let $n = \\dim V$. By the result on the [[dimension of a subspace|0QW084460E]], $W$ is finite-dimensional. Let $k = \\dim W$.\n2. Let $b_1, \\dots, b_k$ be a [[basis|0KCQ9Y3QKA]] of $W$. It is linearly independent in $V$, so it [[extends to a basis|0M4SXE5WW3]] of $V$.\n3. By the definition of [[dimension|03GSZ8RTZJ]], this basis of $V$ has $n$ elements. Write it as $b_1, \\dots, b_n$.\n4. Let $b_1^*, \\dots, b_n^* \\in V^*$ be the [[dual basis|0W6M3PSD08]], so $b_i^*(b_j) = \\delta_{ij}$. Here $\\delta_{ij} = 1$ if $i = j$ and $\\delta_{ij} = 0$ otherwise.\n5. We claim that $b_{k+1}^*, \\dots, b_n^*$ is a basis of the [[annihilator|0R6M158SQ9]] $W^0 = \\set{f \\in V^* : f(w) = 0 \\text{ for all } w \\in W}$.\n6. Each $b_i^*$ with $i \u003e k$ lies in $W^0$. Indeed, each $w \\in W$ is $w = \\sum_{j=1}^{k} c_j b_j$ with $c_j \\in F$. Then $b_i^*(w) = \\sum_{j=1}^{k} c_j \\delta_{ij} = 0$, since $j \\le k \u003c i$.\n7. The forms $b_{k+1}^*, \\dots, b_n^*$ are linearly independent. Suppose $\\sum_{i=k+1}^{n} c_i b_i^* = 0$ with $c_i \\in F$.\n8. Evaluate the sum of step 7 at $b_j$ for $j \u003e k$. This gives $\\sum_{i=k+1}^{n} c_i \\delta_{ij} = c_j = 0$. So all $c_j$ are $0$.\n9. The forms $b_{k+1}^*, \\dots, b_n^*$ span $W^0$. Let $f \\in W^0$ and put $g = \\sum_{i=k+1}^{n} f(b_i) b_i^*$.\n10. For $j \\le k$, we have $g(b_j) = 0$ by step 6, and $f(b_j) = 0$ because $b_j \\in W$.\n11. For $j \u003e k$, we have $g(b_j) = \\sum_{i=k+1}^{n} f(b_i) \\delta_{ij} = f(b_j)$.\n12. So $f$ and $g$ agree on the basis $b_1, \\dots, b_n$. A [[linear map is determined by its values on a basis|0R8X9XA83A]], so $f = g$.\n13. By steps 6 to 12, $b_{k+1}^*, \\dots, b_n^*$ is a basis of $W^0$. So $\\dim W^0 = n - k$.\n14. Hence $\\dim W + \\dim W^0 = k + (n - k) = n = \\dim V$.\n\n## Idea\n\nChoose a basis of $V$ that starts with a basis of $W$. The [[dual basis|0ZD2CF6ZS0]] forms that belong to the remaining basis vectors vanish on $W$, and they give exactly the forms that vanish on $W$. So the annihilator has one dimension for each basis vector outside $W$.\n",
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