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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nA family $(v_1, \\dots, v_k)$ of vectors of $V$ is *linearly independent* when the only [[linear combination|0V8F3E48E6]] of them that equals $0$ is the trivial one. This means that\n$$a_1 v_1 + \\cdots + a_k v_k = 0 \\implies a_1 = \\cdots = a_k = 0$$\nfor all $a_1, \\dots, a_k \\in F$.\nAn arbitrary family $(v_i)_{i \\in I}$ is *linearly independent* when each finite subfamily is linearly independent.\nA family that is not linearly independent is *linearly dependent*.\n\n## Remarks\n\nThe vectors $(1, 0)$ and $(1, 1)$ in $\\R^2$ are linearly independent; the vectors $(1, 2)$ and $(2, 4)$ are linearly dependent.\nA family that contains the zero vector, or the same vector twice, is linearly dependent.\nA family of at least two vectors is linearly dependent if and only if one of its vectors is a linear combination of the others.\nIn $F^n$, a family of more than $n$ vectors is linearly dependent.\n\n## Explanations\n\n### L1 human\n\nVectors are linearly independent when none of them is redundant.\nNo vector can be built from the others, so each one adds a new direction.\nTwo arrows in the plane are independent exactly when they do not lie on one line.\n",
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