Bilinear Form

bilinear form

[¦bī‚lin·ē·ər ′fȯrm] (mathematics) A polynomial of the second degree which is homogeneous of the first degree in each of two sets of variables; thus, it is a sum of terms of the form a_ijx_iy_j, where x₁, … , x_m and y₁, … , y_n are two sets of variables and the a_ij are constants. More generally, a mapping ƒ(x, y) from E × F into R, where R is a commutative ring and E × F is the Cartesian product of two modules E and F over R, such that for each x in E the function which takes y into ƒ(x, y) is linear, and for each y in F the function which takes x into ƒ(x, y) is linear.

Bilinear Form

a form—that is, a homogeneous polynomial—of the second degree from two groups of variables x₁, x₂, . . . , x_n and y₁, y₂, . . . , y_n nof the form

For example, axy is a bilinear form of the variables x and y, and a₁₁x₁y₁ + a₂₁x₂y₁ + a₂₂x₂y₂ is a bilinear form of the variables x₁, x₂, and y₁, y₂. A bilinear form is a particular type of quadratic form.

单词	bilinear form
释义	Bilinear Form bilinear form [¦bī‚lin·ē·ər ′fȯrm] (mathematics) A polynomial of the second degree which is homogeneous of the first degree in each of two sets of variables; thus, it is a sum of terms of the form a_ijx_iy_j, where x₁, … , x_m and y₁, … , y_n are two sets of variables and the a_ij are constants. More generally, a mapping ƒ(x, y) from E × F into R, where R is a commutative ring and E × F is the Cartesian product of two modules E and F over R, such that for each x in E the function which takes y into ƒ(x, y) is linear, and for each y in F the function which takes x into ƒ(x, y) is linear. Bilinear Form a form—that is, a homogeneous polynomial—of the second degree from two groups of variables x₁, x₂, . . . , x_n and y₁, y₂, . . . , y_n nof the form For example, axy is a bilinear form of the variables x and y, and a₁₁x₁y₁ + a₂₁x₂y₁ + a₂₂x₂y₂ is a bilinear form of the variables x₁, x₂, and y₁, y₂. A bilinear form is a particular type of quadratic form.
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