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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.