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  "body": "\n## Summary\n\nThe linear algebra method bounds the size of a combinatorial family by attaching to each member a vector, or a polynomial, in a [[vector space|000AYPA9AJ]].\nOne then proves that these vectors are [[linearly independent|0CRMGPAH83]].\nThe size of the family is then at most the [[dimension|03GSZ8RTZJ]] of the space.\nClassic results proved this way include [[Fisher's inequality|M9XF82GGW8]] and the Oddtown theorem.\nIts polynomial versions, such as the [[combinatorial Nullstellensatz|9ZW7BCJMDT]] and the slice rank argument for the [[cap set problem|1SF8Y9M8YB]], form the polynomial method in combinatorics.\n\n## Explanations\n\n### L1 human\n\nTo show that a family cannot be large, turn each of its members into a vector and show that no vector is a combination of the others.\nThen there can be no more vectors than the dimension of the space.\nA standard example is the Oddtown theorem.\nLet $\\mathcal{F}$ be a family of subsets of $\\set{1, \\ldots, n}$ in which every set has odd size and any two distinct sets meet in an even number of elements.\nOver the [[finite field|0ZXZ82FTX2]] $\\F_2$, the indicator vectors $v_A$ satisfy $v_A \\cdot v_A = 1$ and $v_A \\cdot v_B = 0$ for $A \\ne B$, so they are linearly independent, and $|\\mathcal{F}| \\le n$.\n",
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