V, constitutes a linear transformation T: V→V.
Website owner: James Miller
EXAMPLES OF LINEAR MAPPINGS, LINEAR TRANSFORMATIONS, LINEAR OPERATORS
The following are examples of linear mappings, linear transformations, or linear operators:
1. The mapping y = Ax where A is an mxn matrix, x is an n-vector and y is an m-vector. This represents a linear mapping from n-space into m-space.
2. The mapping y = Ax where A is an nxn matrix, x is an n-vector and y is an n-vector. This represent a linear mapping from n-space into n-space.
3. Let V be the vector space of all n-square matrices over the real field R and A be an n-square matrix over R.. Let v be any member of V. The transformation
w = Av
where w
V, constitutes a linear transformation T: V→V.
4. Let V be the vector space of all polynomials in the variable x over the real field R. Then the
derivative defines a linear transformation D: V→V where, for any polynomial f
V, we let D(f)
= df/dx. For example, D(4x2 + 3x + 5) = 8x + 3.
5. Let V be the vector space of all polynomials in the variable x over the real field R. Then the integral from, say, 0 to 1, defines a linear transformation T: V→R where, for any polynomial
f
V, we let
For example,
.
6. Let V be the vector space of all real-valued continuous functions defined on the interval [0,1].
Then for any f
V the transformation
defines a linear transformation T : V→V.
7. Let V be the vector space of all real-valued continuously differentiable functions defined on
the interval [0,1] and W be the vector space of all real-valued continuous functions defined on
the interval [0,1]. Then for any f
V the transformation
D(f) = f '(x)
defines a linear transformation D:V→W.
8. The integral operator
which associates with function f a certain function g (where k(x, y) is a definite known function called the kernel).
References.
Mathematics, Its Content, Methods and Meaning, III, p.255
Lipschutz. Linear Algebra.
Hoffman and Kunze. Linear Algebra.
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