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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 ...
In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown. Repeat steps 1 and 2 until the system is reduced to a single linear equation.
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.
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
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. However, a random vector may also refer to a random variable that takes its values in a vector space.
In mathematics and physics, vector notation is a commonly used notation for representing vectors, [1] [2] which may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower case, upright boldface type, as in v .
One says that these functions are defined, continuous and differentiable everywhere. This is the case of: All polynomial functions, including constant functions and linear functions; Sine and cosine functions; Exponential function; Some functions are defined everywhere, but not continuous at some points. For example
This example considers the special case where there are exactly two vector and from some real or complex vector space. The vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } are linearly dependent if and only if at least one of the following is true: