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In matematica, una trasformazione lineare (chiamata anche operatore lineare o mappa lineare) è una funzione tra due spazi vettoriali tale che l'operazione di somma di vettori e moltiplicazione per scalari sia preservata. In altre parole preserva le combinazioni lineari.

Nel linguaggio dell'algebra astratta, una trasformazione lineare è un omomorfismo di spazi vettoriali.

Definizione e prime conseguenze

Formalmente, se V e W sono spazi vettoriali sullo stesso campo K, si dice che f : V → W è una trasformazione lineare se per ogni due vettori x e y in V e per ogni scalare a in K, si ha

${\displaystyle f(x+y)=f(x)+f(y)\,}$  (addittività)

${\displaystyle f(ax)=af(x)\,}$                (omomgeneità).

Questo è equivalente al dire che f   "preserva le combinazioni lineari", ovvero per un insieme finito di vettori x₁, ..., x_m e scalari a₁, ..., a_m, si ha :${\displaystyle f(a_{1}x_{1}+\cdots +a_{m}x_{m})=a_{1}f(x_{1})+\cdots +a_{m}f(x_{m}).}$ 

Occasionalmente, V e W possono essere considerati come spazi vettoriali su differenti campi, ed è importante speficicare quale campo è stato utilizzato nella definizione di "lineare". Se V e W sono considerati come spazi sul campo K come sopra, si parla di mappe K-lineari. Per esempio la coniugazione di numeri complessi è una mappa R-lineare C → C, ma non è C-lineare.

Esempi

• La moltiplicazione per una costante è una trasformazione lineare da R a R.

• Se A è una matrice m × n, allora A definisce una trasformazione lineare da Rⁿ a R^m mandando il vettore colonna x ∈ Rⁿ nel vettore colonna Ax ∈ R^m. Ogni trasformazione lineare tra spazi vettoriali finito-dimensionale sono di questo tipo. Si veda la sezione seguente.

• L'integrale è una mappa lineare dallo spazio delle funzioni a valori reali integrabili in qualche intervallo a R

• La derivata è una mappa lineare dallo spazio di tutte le funzioni differenziabili nello spazio di tutte le funzioni.

Examples

• If A is an m × n matrix, then A defines a linear transformation from Rⁿ to R^m by sending the column vector x ∈ Rⁿ to the column vector Ax ∈ R^m. Every linear transformation between finite-dimensional vector spaces arises in this fashion; see the following section.

• The integral yields a linear map from the space of all real-valued integrable functions on some interval to R

• Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.

• If V and W are finite-dimensional vector spaces over the field F, then functions that map linear transformations f : V → W to dim_F(W)-by-dim_F(V) matrices in the way described in the sequel are themselves linear transformations.

Matrices

If V and W are finite-dimensional and bases have been chosen, then every linear transformation from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear transformations: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear transformation Rⁿ → R^m (see Euclidean space).

Let ${\displaystyle \{v_{1},\cdots ,v_{n}\}}$  be a basis for V. Then every vector v in V is uniquely determined by the coefficients ${\displaystyle c_{1},\cdots ,c_{n}}$  in

${\displaystyle c_{1}v_{1}+\cdots +c_{n}v_{n}.}$ 

If f : V → W is a linear transformation,

${\displaystyle f(c_{1}v_{1}+\cdots +c_{n}v_{n})=c_{1}f(v_{1})+\cdots +c_{n}f(v_{n}),}$ 

which implies that the function f is entirely determined by the values of ${\displaystyle f(v_{1}),\cdots ,f(v_{n}).}$ 

Now let ${\displaystyle \{w_{1},\cdots ,w_{m}\}}$  be a basis for W. Then we can represent the values of each ${\displaystyle f(v_{j})}$  as

${\displaystyle f(v_{j})=a_{1j}w_{1}+\cdots +a_{mj}w_{m}.}$ 

So the function f is entirely determined by the values of ${\displaystyle a_{i,j}}$ .

If we put these values into an m-by-n matrix M, then we can conveniently use it to compute the value of f for any vector in V. For if we place the values of ${\displaystyle c_{1},\cdots ,c_{n}}$  in an n-by-1 matrix C, we have MC = f(v).

It should be noted that there can be multiple matrices that represent a single linear transformation. This is because the values of the elements of the matrix depend on the bases that are chosen. Similarly, if we are given a matrix, we also need to know the bases that it uses in order to determine what linear transformation it represents.

Forming new linear transformations from given ones

The composition of linear transformations is linear: if f : V → W and g : W → Z are linear, then so is g o f : V → Z.

If f₁ : V → W and f₂ : V → W are linear, then so is their sum f₁ + f₂ (which is defined by (f₁ + f₂)(x) = f₁(x) + f₂(x)).

If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a (f(x)), is also linear.

In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices, the addition of linear maps corresponds to the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

Endomorphisms and automorphisms

A linear transformation f : V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The identity element of this algebra is the identity map id : V → V.

A bijective endomorphism of V is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V).

If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n by n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by n invertible matrices with entries in K.

Kernel and image

If f : V → W is linear, we define the kernel and the image of f by

${\displaystyle \ker(f)=\{\,x\in V:f(x)=0\,\}}$ 

${\displaystyle \operatorname {im} (f)=\{\,f(x):x\in V\,\}}$ 

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is often useful (but note that it only applies if V is finite dimensional):

${\displaystyle \dim(\ker(f))+\dim(\operatorname {im} (f))=\dim(V)\,}$ 

The number dim(im(f)) is also called the rank of f and written as rk(f). If V and W are finite dimensional, bases have been chosen and f is represented by the matrix A, then the rank of f is equal to the rank of the matrix A. The dimension of the kernel is also known as the nullity of the matrix.

See also

• Transformation matrix
• wikibooks:Algebra:Linear transformations