2.5. Linear mappings

1) F( + ) = F() + F(),

2) F(a) = aF(),

for all , V ja a K.

Proposition 5. If F : Rⁿ -> R^m is a linear mapping, then there exists a unique m x n matrix A such that F() = A for all Rⁿ.

Proof. The existence: Let

_i = F(_i ),  i = 1, ..., n

and

A = (_{1 2} ... _n ).

If Rⁿ, then

= x_{i i}

and

F()	= xi F (i )	=		xi i
	= (1 ...n )		x1		= A .
			:
			xn

The uniqueness: If also B = F(),    Rⁿ, then

A = B,    Rⁿ     => A = B.

It follows from the Proposition 5 that every linear mapping F : Rⁿ -> R^m can be identified with an m x n matrix A, whose ith column is F(_i ), i = 1, ..., n. The matrix A is called the matrix of F relative to the standard bases.

The next proposition, whose proof is omitted, generalizes the previous one:

Proposition 6. Let V and Z be vector spaces, dim V = n, dim Z = m, with bases {₁ , ..., _n } and {₁ , ..., _m }, respectively. If F : V -> Z is a linear mapping, then there exists a unique m x n matrix A such that F() = A for all V. The ith column of A consists of the coordinates of F (_i ) relative to the basis
{₁ , ..., _m }.

The matrix A above is called the matrix of F relative to the bases S_V = {₁ , ..., _n } ja S_Z = {₁ , ..., _m }.

Examples of linear mappings:

1. Matrix multiplication: The mapping F : Rⁿ -> R^m defined by the a matrix A = A_m x n

F () = A   ,       Rⁿ

2. Differentiating defines a linear mapping from the vector space

3. F ( x₁, x₂, x₃ ) = ( x₁ + 2 x₂ - x₃, 2 x₂ + x₃ ), i.e.,

F x1 x2 x3

= x1 + 2 x2 - x3 2 x2 - x3

Example 1: The matrix of the above F relative to the standard bases
Example 2: The matrix of a linear mapping relative to different bases