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This can be concisely written as the matrix inequality , where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. [citation needed] In the above systems both strict and non-strict inequalities may be used. Not all systems of linear inequalities have solutions.
The matrix X on the left is a Vandermonde matrix, whose determinant is known to be () = < (), which is non-zero since the nodes are all distinct. This ensures that the matrix is invertible and the equation has the unique solution A = X − 1 ⋅ Y {\displaystyle A=X^{-1}\cdot Y} ; that is, p ( x ) {\displaystyle p(x)} exists and is unique.
Consider the system of equations + + = + + = + + = The coefficient matrix is = [], and the augmented matrix is (|) = []. Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an infinite number of solutions.
Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution.
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which ...
After days of services honoring former President Jimmy Carter, an official state funeral is set for Thursday at the Washington National Cathedral in Washington, D.C. Carter's state funeral is set ...
The same is true for not less than, . The notation a ≠ b means that a is not equal to b; this inequation sometimes is considered a form of strict inequality. [4] It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set. In engineering sciences, less formal use of the notation is to ...
Amid a string of burglaries targeting NFL players' homes, Miami Dolphins quarterback Tua Tagovailoa is prepared.Tagovailoa revealed in a news conference Wednesday that he has personal security at ...