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 dimension of the function's domain has no relation to the dimension of its range.
Example: Helix
A common example of a vector-valued function is one that depends on a single real parameter t, often representing time, producing a vector v(t) as the result. In terms of the standard unit vectors i, j, k of Cartesian 3-space, these specific types of vector-valued functions are given by expressions such as
r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k {\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }
where f(t), g(t) and h(t) are the coordinate functions of the parameter t, and the domain of this vector-valued function is the intersection of the domains of the functions f, g, and h. It can also be referred to in a different notation:
r ( t ) = ⟨ f ( t ) , g ( t ) , h ( t ) ⟩ {\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }
The vector r(t) has its tail at the origin and its head at the coordinates evaluated by the function. The vector shown in the graph to the right is the evaluation of the function ⟨ 2 cos t , 4 sin t , t ⟩ {\displaystyle \langle 2\cos t,\,4\sin t,\,t\rangle } near t = 19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The helix is the path traced by the tip of the vector as t increases from zero through 8π. In 2D, we can analogously speak about vector-valued functions as:
r ( t ) = f ( t ) i + g ( t ) j {\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} } or
r ( t ) = ⟨ f ( t ) , g ( t ) ⟩ {\displaystyle \mathbf {r} (t)=\langle f(t),g(t)\rangle }
Linear case In the linear case the function can be expressed in terms of matrices:
y = A x , {\displaystyle \mathbf {y} =A\mathbf {x} ,}
where y is an n × 1 output vector, x is a k × 1 vector of inputs, and A is an n × k matrix of parameters. Closely related is the affine case (linear up to a translation) where the function takes the form
y = A x + b , {\displaystyle \mathbf {y} =A\mathbf {x} +\mathbf {b} ,}
where in addition b'' is an n × 1 vector of parameters. The linear case arises often, for example in multiple regression, where for instance the n × 1 vector y ^ {\displaystyle {\hat {y}}} of predicted values of a dependent variable is expressed linearly in terms of a k × 1 vector β ^ {\displaystyle {\hat {\boldsymbol {\beta }}}} (k < n) of estimated values of model parameters:
y ^ = X β ^ , {\displaystyle {\hat {\mathbf {y} }}=X{\hat {\boldsymbol {\beta }}},}
in which X (playing the role of A in the previous generic form) is an n × k matrix of fixed (empirically based) numbers.
Parametric representation of a surface A surface is a 2-dimensional set of points embedded in (most commonly) 3-dimensional space. One way to represent a surface is with parametric equations, in which two parameters s and t determine the three Cartesian coordinates of any point on the surface:
( x , y , z ) = ( f ( s , t ) , g ( s , t ) , h ( s , t ) ) ≡ F ( s , t ) . {\displaystyle (x,y,z)=(f(s,t),g(s,t),h(s,t))\equiv \mathbf {F} (s,t).}
Here F is a vector-valued function. For a surface embedded in n-dimensional space, one similarly has the representation
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