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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.\nLet $v_1, \\dots, v_k \\in V$.\nA *linear combination* of $v_1, \\dots, v_k$ is a vector of the form\n$$a_1 v_1 + a_2 v_2 + \\cdots + a_k v_k,$$\nwhere $a_1, \\dots, a_k \\in F$ are the *coefficients*.\nFor an arbitrary family $(v_i)_{i \\in I}$, a linear combination is a sum $\\sum_{i \\in I} a_i v_i$ in which only finitely many coefficients $a_i$ are nonzero.\n\n## Remarks\n\nThe empty linear combination, with $k = 0$, is the zero vector.\nThe vector $(3, 5)$ is the linear combination $3 e_1 + 5 e_2$ of the standard basis vectors of $F^2$.\nThe set of all linear combinations of a family is its [[linear span|0DTY8B9HMS]].\nThe [[matrix–vector product|0PDVQEP0ZC]] $Ax$ is a linear combination of the columns of $A$, with the entries of $x$ as coefficients.\n\n## Explanations\n\n### L1 human\n\nA linear combination mixes some vectors: scale each one by a number, then add the results.\nWith the vectors $(1, 0)$ and $(0, 1)$ we can make every vector of the plane in this way.\nInfinite sums are not allowed: only finitely many coefficients can be nonzero.\n",
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