Linear mapping, linear transformation, homomorphism, isomorphism, operators, linear and nonlinear transformations, change of basis, similar matrices

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Def. Linear mapping (or linear transformation). A linear mapping (or linear transformation) is a mapping defined on a vector space that is linear in the following sense: Let V and W be vector spaces over the same field F. A linear mapping is a mapping V → W which takes ax + by into ax' + by' for all a and b if it takes vectors x and y in V into x' and y' in W. The numbers a and b may be real, complex or of any field for which multiplication with elements of the field is defined.

A linear mapping is a mapping that "preserves" the two basic operations of a vector space, that of vector addition and that of scalar multiplication. A linear mapping effects a vector space homomorphism. If it is nonsingular it effects a vector space isomorphism. A consequence of the linearity of a mapping is that subspaces are mapped into subspaces.

An example of such a mapping in n-dimensional space, V_n(F), is

y = Ax

where x and y are n-vectors and A is an nxn matrix. This represents a mapping of n-space into itself.

Note. There are alternative but equivalent definitions of a linear mapping. Another definition is: A linear mapping is a mapping Q: V → W that satisfies the following two conditions:

1. For any x,y in V, Q(x + y) = Q(x) + Q(y)

2. For any a in F and any x in V, Q(ax) = aQ(x)

Synonyms: Linear Transformation, Linear Operator, Linear map, Vector Space Homomorphism, Linear vector function, Morphism, Homogeneous affine transformation

Isomorphism. Two axiomatically-defined abstract mathematical systems as, for example, two groups, rings, linear spaces , etc. are said to be isomorphic to each other if they are equivalent structurally, algebraically; in their internal workings, with like elements corresponding in a one-to-one fashion; the differences between them being only superficial ones as in the names we give the elements and the way we denote the law of combination.

Homomorphism. As with an isomorphism, a homomorphism is also a correspondence between two mathematical structures that are structurally, algebraically identical. However, there is an important difference between a homomorphism and an isomorphism. An isomorphism is a one-to-one mapping of one mathematical structure onto another. A homomorphism is a many-to-one mapping of one structure onto another. Where an isomorphism maps one element into another element, a homomorphism maps a set of elements into a single element. Its definition sounds much the same as that for an isomorphism but allows for the possibility of a many-to-one mapping. An isomorphism is actually a special case of a homomorphism.

Concept of an operator. The term “operator” is another term for function. An operator assigns an object from one set (the range) to an object from another set (the domain). If we are talking about vector spaces we think of an operator as “operating” on one vector to produce another vector. It is viewed as a black box that operates on vectors to produce other vectors. The black box has an input and an output. We input a vector into the box and it then outputs a vector. An example is the matrix A in the matrix equation y = Ax where A is viewed as a black box that operates on the vector x to produce vector y. Here matrix A maps a vector x from one space (the domain) into the vector y in another space (the range).

Linear transformation y = Ax. The linear transformation of primary interest in matrix theory is the transformation y =Ax. If A is an mxn matrix then A can be viewed as a linear operator that maps n-vectors of n-space into m-vectors of m-space. If A is an n-square matrix it can be viewed as mapping n-vectors into n-vectors i.e. mapping n-space into itself. Matrix A may be singular or non-singular. The mapping has an inverse if and only if the matrix A is non-singular. The inputs that matrix A operates on can be viewed as vectors or as points. It is just a matter of point of view, of outlook, of terminology. We can think of matrix A as mapping figures in n-space into other figures. When we think of y = Ax as mapping points into other points we may call it a linear point transformation — just to emphasize that point of view.

For the case of point transformations of real n-space into itself it is natural to ask what kind of transformation y = Ax effects on figures. What effect does it have on circles in 2-space, spheres in 3-space, etc.? In fact, provided the determinant |A| 0, it is a special kind of affine transformation called a homogeneous affine transformation. Such transformations carry points into points, straight lines into straight lines, parallel lines into parallel lines, and if a point divides a line segment into a given ratio, the image of the point divides the image of the segment into the same ratio. Every curve or surface of second degree is carried into another of the second degree and, in general, every curve or surface of degree n is carried into another of the same degree. A region of space is carried into an image region whose volume is a constant factor (namely, Δ, the determinant of the transformation) times the volume of the original region. A homogeneous affine transformation in three dimensional space produces an effect that is equivalent to some rotation of the coordinate system about the origin and then three contractions/elongations perpendicular to the three coordinate planes with certain coefficients k₁, k₂, k₃.

Images of the elementary unit vectors under a linear transformation. Consider the following question: What are the images of the elementary unit vectors

E₁ = [1, 0, 0, ...., 0]^T

E₂ = [0, 1, 0, ...., 0]^T

..............................

E_n = [0, 0, 0, ...., 1]^T

under a linear transformation Y = AX? Answer: The image of the elementary unit vector E_i is the i-th column of matrix A i.e. AE_i = (i-th column of A).

***********************

Theorems.

Theorem 1. A linear transformation Y = AX is uniquely determined when the images of the elementary unit basis vectors

E₁ = [1, 0, 0, ...., 0]^T

E₂ = [0, 1, 0, ...., 0]^T

..............................

E_n = [0, 0, 0, ...., 1]^T

are known. The images of these vectors are the respective columns of A (the respective images of the vectors define the columns of the transformation matrix A).

Singular and nonsingular transformations. A linear transformation Y = AX is called nonsingular if the images of distinct vectors X_iare distinct vectors Y_i . Otherwise the transformation is called singular.

Theorem 2. A linear transformation Y = AX is nonsingular if and only if A, the matrix of the transformation, is nonsingular.

Theorem 3. A nonsingular linear transformation carries linearly independent vectors into linearly independent vectors and linearly dependent vectors into linearly dependent vectors.

Theorem 4. Under a nonsingular linear transformation Y = AX the image of an r-dimensional vector space is an r-dimensional vector space i.e. an r-dimensional space is mapped onto an r-dimensional space and not mapped onto some space of a lower dimension. For example, a three-dimensional space is mapped into a three-dimensional space and not into some two-dimensional or one-dimensional space, as can happen with singular transformations. In particular, a nonsingular transformation maps V_n(F) onto itself.

Theorem 5. The elementary vectors E_i of V_n may be transformed into any set of n linearly independent n-vectors by a nonsingular linear transformation and conversely.

Theorem 6. Suppose Y = AX carries a vector X into a vector Y, Z = BY carries Y into Z and

W = CZ carries Z into W. Then W = CBAX carries X into W.

Theorem 7. Given any two sets of n linearly independent n-vectors, there exists a nonsingular linear transformation which carries the vectors of one set into the vectors of the other set.

Changes in the expression of a linear transformation due to a change in basis. Let Y_Z = AX_Z be a linear transformation expressed with respect to some original Z-basis { Z₁, Z₂, ... ,Z_n }. What is the expression for this same transformation when expressed with respect to some arbitrary other W-basis?

We know from a previous theorem that

X_Z = Z^-1WX_W and Y_Z = Z^-1WY_W

Substituting into Y_Z = AX_Z we get

Z^-1WY_W = AZ^-1WX_W

Y_W = (Z^-1W)^-1A(Z^-1W)X_W

Let Q = Z^-1W

So we have the final result:

Y_W = Q ^-1AQX_W

Similar matrices. Two matrices A and B such that there exists a nonsingular matrix Q for which B = Q^-1AQ are called similar.

Theorem 8. If Y_Z = AX_Z is a linear transformation of V_n relative to a given basis (Z-basis) and

Y_W = BX_W is the same linear transformation relative to another basis (W-basis), then A and B are similar.

References.

Ayres. Matrices. (Schaum).

Lipschutz. Linear Algebra. (Schaum)

Klein. Geometry. Elementary Mathematics from an Advanced Standpoint.