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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nA *linear form* on $V$ is a [[linear map|0NRYHJPQG2]] $f \\colon V \\to F$, where $F$ is a vector space over itself.\nThus $f(u + v) = f(u) + f(v)$ and $f(av) = a f(v)$ for all $u, v \\in V$ and $a \\in F$.\n\n## Remarks\n\nA linear form is also called a *linear functional* or a *covector*.\nOn $F^n$, each linear form is $x \\mapsto c_1 x_1 + \\cdots + c_n x_n$ for unique scalars $c_1, \\dots, c_n$.\nThe [[trace|0XC1RDBY7A]] is a linear form on the space of $n \\times n$ matrices.\nThe linear forms on $V$ form the [[dual space|07JBSZE0V4]] of $V$.\n\n## Explanations\n\n### L1 human\n\nA linear form takes a vector and returns one number, in a linear way.\nOn the plane, the map $(x, y) \\mapsto 2x - 3y$ is a linear form.\nIts kernel is the line $2x = 3y$, unless the form is zero.\n",
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