Linear transformation

Categories: Abstract algebra | Linear algebra

In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, it "preserves linear combinations".

In the language of abstract algebra, a linear transformation is a homomorphism of vector spaces.

Contents 1 Definition and first consequences 2 Examples 3 Matrices 4 Examples of linear transformation matrices 5 Forming new linear transformations from given ones 6 Endomorphisms and automorphisms 7 Kernel and image 8 Continuity 9 See also

Definition and first consequences

Formally, if V and W are vector spaces over the same ground field K, we say that f : V → W is a linear transformation if for any two vectors x and y in V and any scalar a in K, we have

<math>f(x+y)=f(x)+f(y) \,</math> (additivity)

<math>f(ax)=af(x) \,</math>               (homogeneity).

This is equivalent to saying that f   "preserves linear combinations", i.e., for any vectors x₁, ..., x_m and scalars a₁, ..., a_m, we have

<math>f(a_1 x_1+\cdots+a_m x_m)=a_1 f(x_1)+\cdots+a_m f(x_m).</math>

Occasionally, V and W can be considered as vector spaces over different ground fields, and it is then important to specify which field was used for the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear.

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 <math>\{v_1, \cdots, v_n\}</math> be a basis for V. Then every vector v in V is uniquely determined by the coefficients <math>c_1, \cdots, c_n</math> in

<math>c_1 v_1+\cdots+c_n v_n.</math>

If f : V → W is a linear transformation,

<math>f(c_1 v_1+\cdots+c_n v_n)=c_1 f(v_1)+\cdots+c_n f(v_n),</math>

which implies that the function f is entirely determined by the values of <math>f(v_1),\cdots,f(v_n).</math>

Now let <math>\{w_1, \cdots, w_m\}</math> be a basis for W. Then we can represent the values of each <math>f(v_j)</math> as

<math>f(v_j)=a_{1j} w_1 + \cdots + a_{mj} w_m.</math>

So the function f is entirely determined by the values of <math>a_{i,j}</math>.

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 <math>c_1, \cdots, c_n</math> 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.

Examples of linear transformation matrices

Some special cases of linear transformations of two-dimensional space R² are illuminating:

• rotations: no real eigenvectors (complex eigenvalue/eigenvector pairs exist). Example (rotation by 90 degrees counterclockwise):

  <math>A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}.</math>

• reflection: eigenvectors are perpendicular and parallel to the line of symmetry. The eigenvalues are -1 and 1, respectively. Example (symmetry against the x axis):

  <math>A=\begin{bmatrix}1 & 0\\ 0 & -1\end{bmatrix}.</math>

• scaling:
  • uniform scaling: all vectors are eigenvectors, and the eigenvalue is the scale factor. Example (scale by 2 in all directions):

  <math>A=\begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}.</math>

  • directional scaling: eigenvalues are the scale factor and 1
  • directionally differential scaling: eigenvalues are the scale factors
• projection onto a line: vectors on the line are eigenvectors with eigenvalue 1 and vectors in the direction of projection (which may or may not be perpendicular) are eigenvectors with eigenvalue 0. Example:

  <math>A=\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}.</math>

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.

Thus the set L(V,W) of linear maps from V to W forms a vector space over K itself. Furthermore, in the case that V=W, this vector space is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and composition of maps is always associative. This case is discussed in more detail below.

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

<math>\ker(f)=\{\,x\in V:f(x)=0\,\}</math>

<math>\operatorname{im}(f)=\{\,f(x):x\in V\,\}</math>

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is often useful:

<math>

 \dim(\ker( f )) 

+ \dim(\operatorname{im}( f )) = \dim( V ) \,</math>

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.

Continuity

A linear transformation between normed spaces is continuous iff it is a bounded operator. If its domain is finite-dimensional, this is always the case. An example of an unbounded, hence not continuous, linear transformation is differentiation, with the maximum norm (a function with small values can have a derivative with large values).

See also

• Transformation matrix
• Continuous linear operator
• wikibooks:Algebra:Linear transformationsde:Lineare Abbildung

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