"linear form" meaning in English

{
  "forms": [
    {
      "form": "linear forms",
      "tags": [
        "plural"
      ]
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  "lang_code": "en",
  "pos": "noun",
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          "name": "Functions",
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          "name": "Linear algebra",
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        }
      ],
      "derived": [
        {
          "word": "linear form in logarithms"
        }
      ],
      "examples": [
        {
          "text": "In #x5C;mathbb#x7B;R#x7D;ⁿ, if vectors are represented as column vectors, then linear forms are represented as row vectors, and their action on vectors is given by the matrix product, with the row vector on the left and the column vector on the right.",
          "type": "example"
        },
        {
          "text": "1961 [Prentice-Hall], Richard A. Silverman (translator), Georgi E. Shilov, An Introduction to the Theory of Linear Spaces, 1974, Dover, page 66,\nA more general linear form in the same space is the expression\nf(x)=∑ₖ₊₁ⁿc_kζₖ\nwith arbitrary fixed coefficients c_1,c_2…,c_n."
        },
        {
          "ref": "1998, F. G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, 2nd edition, Cambridge University Press, page 2:",
          "text": "In the theory of distributions, functions are replaced by linear forms on an auxiliary vector space, whose members are called test functions.",
          "type": "quote"
        },
        {
          "ref": "2005, Martin Kreuzer, Lorenzo Robbiano, Computational Commutative Algebra 2, Springer, page 264:",
          "text": "In order to better understand the effect of reducing a K-vector subspace V#x5C;subseteqP#x5F;d modulo a generic linear form, we generalize Proposition 5.5.18 as follows.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A linear functional."
      ],
      "id": "en-linear_form-en-noun-4lfpOkDS",
      "links": [
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      "raw_glosses": [
        "(linear algebra) A linear functional."
      ],
      "related": [
        {
          "word": "bilinear form"
        },
        {
          "word": "multilinear form"
        },
        {
          "word": "differential form"
        },
        {
          "word": "polynomial form"
        },
        {
          "word": "dual space"
        }
      ],
      "synonyms": [
        {
          "sense": "linear functional",
          "word": "covector"
        },
        {
          "sense": "linear functional",
          "word": "linear functional"
        },
        {
          "sense": "linear functional",
          "word": "one-form"
        }
      ],
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        "mathematics",
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      "wikipedia": [
        "linear form"
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  "word": "linear form"
}

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  "derived": [
    {
      "word": "linear form in logarithms"
    }
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  "forms": [
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      "form": "linear forms",
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      "word": "differential form"
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    {
      "word": "polynomial form"
    },
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      "word": "dual space"
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      "examples": [
        {
          "text": "In #x5C;mathbb#x7B;R#x7D;ⁿ, if vectors are represented as column vectors, then linear forms are represented as row vectors, and their action on vectors is given by the matrix product, with the row vector on the left and the column vector on the right.",
          "type": "example"
        },
        {
          "text": "1961 [Prentice-Hall], Richard A. Silverman (translator), Georgi E. Shilov, An Introduction to the Theory of Linear Spaces, 1974, Dover, page 66,\nA more general linear form in the same space is the expression\nf(x)=∑ₖ₊₁ⁿc_kζₖ\nwith arbitrary fixed coefficients c_1,c_2…,c_n."
        },
        {
          "ref": "1998, F. G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, 2nd edition, Cambridge University Press, page 2:",
          "text": "In the theory of distributions, functions are replaced by linear forms on an auxiliary vector space, whose members are called test functions.",
          "type": "quote"
        },
        {
          "ref": "2005, Martin Kreuzer, Lorenzo Robbiano, Computational Commutative Algebra 2, Springer, page 264:",
          "text": "In order to better understand the effect of reducing a K-vector subspace V#x5C;subseteqP#x5F;d modulo a generic linear form, we generalize Proposition 5.5.18 as follows.",
          "type": "quote"
        }
      ],
      "glosses": [
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      "raw_glosses": [
        "(linear algebra) A linear functional."
      ],
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  "synonyms": [
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    {
      "sense": "linear functional",
      "word": "linear functional"
    },
    {
      "sense": "linear functional",
      "word": "one-form"
    }
  ],
  "word": "linear form"
}