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Linear Transformation:

Linear Transformation:

Definition of linear transformation:

A linear transformation L of the vector space V into the vector space W is a function (denoted by ) such that for ,

(a) .

(b) .

Note:

If and L is a linear transformation, L is also called a linear operator on V.

Note:

is called the image of .

Example:

Let

.

A linear transformation L of (V) into (W) is a function such that

(a) , where

and .

(b) where .

Several special cases of the above linear transformation are the following:

1. Projection: is defined by

.

L is a linear transformation since

(a) for any ,

.

(b) for ,

.

Dilation: is defined by

.

Constraction: is defined by

Both and are linear transformations.

3.

Let and . Then, .

.

.

Rotation: is defined by

.

L is a linear transformation.

4. Let A be fixed matrix. Then,

defined by

is a linear transformation since

(a) for any ,

.

(b) for ,

.

Example:

Let

,

where is the set of all the polynomials of degrees . Is L a linear transformation?

[solution:]

L is a linear transformation since

(a) for any in ,

.

(b) for ,

Example:

Let , L is the operation of taking the derivative, for example,

.

Is L a linear transformation?

[solution:]

L is a linear transformation since

(a) for any in ,

.

(b) for ,

Example:

is defined by

.

Is L a linear transformation?

[solution:]

L is not a linear transformation since

(a) for any ,

Important result:

Let be a linear transformation. Then,

• , where is the zero vector in V and is the zero vector in W.

• .

• For any vectors in V and any scalars , then

.

• If V is an n-dimensional vector space and be a basis for V. If is any vector in V, then is a linear combination of .

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Tags: linear transformation:, a linear, linear, transformation, definition

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