英语词汇“linearly independent”的英英意思、用法、释义、翻译、读音、例句、短语、词组-英语词典

d. Of one of a set of equations, axioms, or quantities in respect of the others: incapable of being expressed in terms of, or of being derived or deduced from, the others; hence applied to a set of axioms, etc., all of which have this property; linearly independent, (of each of a set of equations or quantities) incapable of being expressed as a linear combination of the others, i.e. satisfying no relation of the form a₁x₁+ a₂x₂+ …+ a_nx_n = o (where x_i are the quantities and a_i arbitrary constants) unless a₁ = a₂ =…= a_n = o.

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the world > relative properties > number > mathematics > [adjective] > relating to mathematical enquiry

canonical1738

independenta1739

algorithmic1799

irreflexive1890

uninterpreted1898

consistent1928

polynomial time1972

the world > relative properties > number > algebra > [adjective] > relating to expressions > relating to equations > systems of

holonomous1899

consistent1902

holonomic1904

linearly independent1959

a1739 N. Saunderson Elements Algebra (1740) I. ii. 105 If a problem be justly proposed, it ought to have as many independent conditions..as there are unknown quantities to be discovered by them.

1798 J. Wood Elem. Algebra (ed. 2) 73 These equations must also be independent, that is, not deducible one from another.

1875 Encycl. Brit. I. 541/1 A problem is limited when the conditions furnish just as many independent equations as there are unknown quantities to be determined: if there be fewer, the problem is indeterminate; but if there be more, the problem in general admits of no solution.

1885 A. R. Forsyth Treat. Differential Equations v. 152 We have now obtained the complete integral of Legendre's equation in all cases when n is a real constant, by deducing two integrals which are linearly independent..of one another.

1902 Trans. Amer. Math. Soc. 3 142 Hilbert states..that his body of axioms consists of independent axioms, that is, that no one of the axioms is logically deducible from the remaining axioms.

1931 L. J. Rouse College Algebra v. 69 The equations 2x + y = 5 and x − y = 4 cannot be reduced to the same form and are therefore independent.

1941 O. Helmer tr. A. Tarski Introd. Logic §39. 131 We strive to arrive at an axiom system which does not contain a single superfluous statement..which can be derived from the remaining axioms... An axiom system of this kind is called independent (or a system of mutually independent axioms).

1944 A. Church Introd. Math. Logic I. i. 25 An axiom of a logistic system is said to be independent if, in the system whose axioms and rules consist of all axioms and rules of the original system except that one, the suppressed axiom is not a theorem.

1959 G. James & R. C. James Math. Dict. (ed. 2) 107/1 The numbers 3 and π are linearly independent with respect to rational numbers, since a₁·3 + a₂·π can not be zero if a₁ and a₂ are rational numbers, not both zero. Since −1·3 + (3/π)π = 0, 3 and π are linearly dependent with respect to real numbers.

1961 J. L. Powell & B. Crasemann Quantum Mech. v. 117 Two solutions of Eq. (5–47) (or, more generally, any two functions of x) are linearly independent if the equation C₁ψ₁ + C₂ψ₂ = 0 cannot be satisfied identically in x for any choice of the constants C₁ and C₂ except C₁ = C₂ = 0.

1961 J. L. Powell & B. Crasemann Quantum Mech. v. 118 To linearly independent solutions, ψ₁ and ψ₂, are a complete set in the sense that every solution of Eq. (5–47) can be expressed as a linear combination of ψ₁ and ψ₂.

1965 G. E. Hughes & D. G. Londey Elements Formal Logic xviii. 132 Since A₄ is non-independent, the axiom set for PM could be reduced by one. But no further reduction of this sort is possible; neither A1 nor A2 nor A3 nor A5 is a consequence of the other three under Substitution and Detachment, and these four are therefore said to be independent axioms.

单词	linearly independent
释义	> as lemmas linearly independent d. Of one of a set of equations, axioms, or quantities in respect of the others: incapable of being expressed in terms of, or of being derived or deduced from, the others; hence applied to a set of axioms, etc., all of which have this property; linearly independent, (of each of a set of equations or quantities) incapable of being expressed as a linear combination of the others, i.e. satisfying no relation of the form a₁x₁+ a₂x₂+ …+ a_nx_n = o (where x_i are the quantities and a_i arbitrary constants) unless a₁ = a₂ =…= a_n = o. ΘΚΠ the world > relative properties > number > mathematics > [adjective] > relating to mathematical enquiry canonical1738 independenta1739 algorithmic1799 irreflexive1890 uninterpreted1898 consistent1928 polynomial time1972 the world > relative properties > number > algebra > [adjective] > relating to expressions > relating to equations > systems of holonomous1899 consistent1902 holonomic1904 linearly independent1959 a1739 N. Saunderson Elements Algebra (1740) I. ii. 105 If a problem be justly proposed, it ought to have as many independent conditions..as there are unknown quantities to be discovered by them. 1798 J. Wood Elem. Algebra (ed. 2) 73 These equations must also be independent, that is, not deducible one from another. 1875 Encycl. Brit. I. 541/1 A problem is limited when the conditions furnish just as many independent equations as there are unknown quantities to be determined: if there be fewer, the problem is indeterminate; but if there be more, the problem in general admits of no solution. 1885 A. R. Forsyth Treat. Differential Equations v. 152 We have now obtained the complete integral of Legendre's equation in all cases when n is a real constant, by deducing two integrals which are linearly independent..of one another. 1902 Trans. Amer. Math. Soc. 3 142 Hilbert states..that his body of axioms consists of independent axioms, that is, that no one of the axioms is logically deducible from the remaining axioms. 1931 L. J. Rouse College Algebra v. 69 The equations 2x + y = 5 and x − y = 4 cannot be reduced to the same form and are therefore independent. 1941 O. Helmer tr. A. Tarski Introd. Logic §39. 131 We strive to arrive at an axiom system which does not contain a single superfluous statement..which can be derived from the remaining axioms... An axiom system of this kind is called independent (or a system of mutually independent axioms). 1944 A. Church Introd. Math. Logic I. i. 25 An axiom of a logistic system is said to be independent if, in the system whose axioms and rules consist of all axioms and rules of the original system except that one, the suppressed axiom is not a theorem. 1959 G. James & R. C. James Math. Dict. (ed. 2) 107/1 The numbers 3 and π are linearly independent with respect to rational numbers, since a₁·3 + a₂·π can not be zero if a₁ and a₂ are rational numbers, not both zero. Since −1·3 + (3/π)π = 0, 3 and π are linearly dependent with respect to real numbers. 1961 J. L. Powell & B. Crasemann Quantum Mech. v. 117 Two solutions of Eq. (5–47) (or, more generally, any two functions of x) are linearly independent if the equation C₁ψ₁ + C₂ψ₂ = 0 cannot be satisfied identically in x for any choice of the constants C₁ and C₂ except C₁ = C₂ = 0. 1961 J. L. Powell & B. Crasemann Quantum Mech. v. 118 To linearly independent solutions, ψ₁ and ψ₂, are a complete set in the sense that every solution of Eq. (5–47) can be expressed as a linear combination of ψ₁ and ψ₂. 1965 G. E. Hughes & D. G. Londey Elements Formal Logic xviii. 132 Since A₄ is non-independent, the axiom set for PM could be reduced by one. But no further reduction of this sort is possible; neither A1 nor A2 nor A3 nor A5 is a consequence of the other three under Substitution and Detachment, and these four are therefore said to be independent axioms. extracted from independentadj.n. < as lemmas
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> as lemmas