“non-homogeneous set of linear equations”的概念、定义、翻译、参考文献-科学参考

non-homogeneous set of linear equations

Mathematics

A set of m linear equations in n unknowns x₁, x₂,…, x_n that has the form

$\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} & = b_{1}, \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} & = b_{2}, \\ ⋮ \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} & = b_{m}, \end{array}$

where b₁, b₂,…, b_m are not all zero. (Compare with homogeneous set of linear equations.) In matrix notation, this set of equations may be written as Ax = b, where A is the m × n matrix [a_ij], and b ≠ 0 and x are column vectors:

$b = [\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{matrix}], x = [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{matrix}] .$

Such a set of equations may be inconsistent, have a unique solution, or have infinitely many solutions (see simultaneous linear equations). If m = n, A is square and the set of equations has a unique solution, namely x = A⁻¹b, if and only if A is invertible.

单词	non-homogeneous set of linear equations
释义	non-homogeneous set of linear equations Mathematics A set of m linear equations in n unknowns x₁, x₂,…, x_n that has the form $\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} & = b_{1}, \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} & = b_{2}, \\ ⋮ \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} & = b_{m}, \end{array}$ where b₁, b₂,…, b_m are not all zero. (Compare with homogeneous set of linear equations.) In matrix notation, this set of equations may be written as Ax = b, where A is the m × n matrix [a_ij], and b ≠ 0 and x are column vectors: $b = [\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{matrix}], x = [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{matrix}] .$ Such a set of equations may be inconsistent, have a unique solution, or have infinitely many solutions (see simultaneous linear equations). If m = n, A is square and the set of equations has a unique solution, namely x = A⁻¹b, if and only if A is invertible.
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