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A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the ...
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding section.
For example, the set of all vectors (x, y, z) (over real or rational numbers) satisfying the equations + + = + = is a one-dimensional subspace. More generally, that is to say that given a set of n independent functions, the dimension of the subspace in K k will be the dimension of the null set of A , the composite matrix of the n functions.
Interval vector, in musical set theory, an array that expresses the intervallic content of a pitch-class set; Probability vector, in statistics, a vector with non-negative entries that sum to one. Random vector or multivariate random variable, in statistics, a set of real-valued random variables that may be correlated.
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns ...
Such an equivalence class is called a vector, more precisely, a Euclidean vector. [13] The equivalence class of (A, B) is often denoted . A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment (A, B)) and same direction (e.g., the direction from A to B). [14]
A quadratic equation is one which includes a term with an exponent of 2, for example, , [40] and no term with higher exponent. The name derives from the Latin quadrus , meaning square. [ 41 ] In general, a quadratic equation can be expressed in the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} , [ 42 ] where a is not zero (if it were ...
In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When the function is of only one variable, it is of the form = +, where a and b are constants, often real numbers.