Let V and W be vector spaces over F, and let T:VmapsW be a linear transformation. Then: (1) dimF(T(V)) < dimF(V) and (2) If T is an isomorphism, dimF(T(V)) = dimF(V). Let V be a vector space. If V has a finite set of generators, then we say that V is finite-dimensional and we define its dimension, denoted dimFV, to be the minimum possible number of elements in such a set. If V does not have a finite set of generators, then we say the V is infinite-dimensional, and we define dimFV to be infinity Let W subset of V. Then W is of subspace of V is equivalent to For all w,w' element of W, alpha element of F, we have w - w' element of W, alphaw element of W. If f:GmapsH is a homomorphism. The kernel of f is ker(f) = {x element of G | f(x) = 1H}. Let G1 and G2 be groups. An isomorphism from G1 to G2 is an injection phi:G1mapsG2 such that phi(a · b) = phi(a) · phi(b)  (a,b element ofG1). Let f :GmapsH be a homomorphism. Then f is an injection equivalent ker(f) = {1G}. Let G and H be groups. A function f :GmapsH which satisfies f(g · g') = f(g) · f(g') for all g, g' element of G is called a homomorphism of G into H. The set of real numbers. A function f : AmapsB is said to be surjective (or onto) if for every y element of B there exists x element of A such that f(x) = y.
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Linear Transformations

In this section we will again pursue the analogy between vector spaces, on the one hand, and groups and rings on the other, by defining an analogue of the homomorphisms of group theory and ring theory. In the case of vector spaces, the homomorphisms are called linear transformations (or linear operators).

Definition 1: Let V and W be vector spaces over the same field F. A linear transformation from V to W is a function T:VmapsW such that for all v,v' element of V and all alpha element of F, we have

(1)
T(v + v') = T(v) + T(v'),
(2)
T(alphav) = alphaT(v).

Note that a linear transformation T:mapsW is a group homomorphism from the additive group of V to the additive group of W. Thus, in particular, if v,v' element of V, then

(3)
T(v - v') = T(v) - T(v').

Let v1,...,vn element of V, alpha1,...,alphan element of F. By repeated application of (1) and (2), we see that if T:mapsW is a linear transformation, then

(4)
T(alpha1v1 + ... + alphavn) = alpha1T(v1) + ... + alphanT(vn).

Example 1: Let alpha1,...,alphan element of F. Then the mapping T:FnmapsF1 defined by T((a1,...,an)) = alpha1a1 + ... + alphanan is a linear transformation.

Example 2: Let V and W be finite-dimensional vector spaces over the field F and let {e1,...,en}, {f1,...,fm} be bases for V and W, respectively. Let T:mapsW be a linear transformation. We may describe the effect of T as follows: Since T(ei) element of W, we have

T(ei) = a1if1 + ... + amifm   (1 < i < n),

where aij element of F. The set {aij}, which consists of mn elements of F, completely determines T, For if v element of V, then

v = alpha1e1 + ... + alphanen,   alphai element of F.

Therefore, by (4),

T(v) = alpha1T(e1) + ... + alphanT(en)
(5)
= alpha1(a11f1 + ... + am1fm)+ ... +alphan(a1nf1 + ... + amnfm)
= (a11alpha1 + a1nalphan)f1 + ... + (am1alpha1 + ... + amnalphan)fn.

Thus, the image of any vector v with respect to T is completely determined by the set {aij}. Conversely, given any set with {aij} of mn elements of F, formula (5) defines a linear transformation of V into W.

Example 3: Let V = W = F[X], considered as a vector space over F. If

f = a0 + a1X + ... + anXn element of F[X],

let us define the formal derivative of f, denoted Df, by

Df = 1 · a1 + ... + n · anXn-1.

Then D(f + g) = Df + Dg and D(alphaf) = alphaDf for all f,g element of F[X], alpha element of F. Thus, D:F[X]mapsF[X] is a linear transformation.

Example 4: Let C[0,1] = the vector space over R of all continuous functions f:[0,1]mapsR. If f element of C[0,1], set

T(f ) = integral from 0 to 1 f(x)dx.

Then, by the properties if the integral proved in calculus, we have

T(f + g) = integral from 0 to 1 ( f(x) + g(x))dx
= integral from 0 to 1 f(x)dx + integral from 0 to 1g(x)dx
= T(f ) + T(g),
T(alphaf ) = integral from 0 to 1alphaf(x)dx
= alphaintegral from 0 to 1 f(x)dx
= alphaT(f )

for all f,g element of C[0,1], alpha element of R. Thus, T:C[0,1]mapsR1 is a linear transformation.

As in group theory, and ring theory let us define the kernel of the linear transformation T:intregal from 0 to 1mapsW, denoted ker(T), by

(6)
ker(T) = {v element of V | T(v) = 0}.

Proposition 2: Let T:VmapsW be a linear transformation.

(1) T(V) is a subspace of W.

(2) ker(T) is a subspace of V.

Proof: (1) Let w,w' element of T(V), alpha element of F. Then there exist v,v' element of V such that w = T(v), w' = T(v'). Then, by (3),

w - w' = T(v) - T(v') = T(v - v') element of T(V).

since v - v' element of V. Similarly, by (2),

alphaw = alphaT(v) = T(alphav) element of T(V),

since alphav element of V. Thus, by Proposition 2 of the section on subspaces, T(V) is a subspace if W. (2) Let v,v' element of ker(T), alpha element of F. Then by (3) and the fact that v,v' element of ker(T),

T(v - v') = T(v) - T(v')
= 0 - 0
= 0
implies v - v' element of ker(T).

And, by (2),

T(alphav) = alphaT(v) = alpha0 = 0
implies alphav element of ker(T).

Thus, ker(T) is a subspace of V.

Let T:mapsW be a linear transformation. Since T is a homomorphism of additive groups, we have, by Corollary 5 of the section on group homomorphisms, the following result.

Proposition 3: T is an isomorphism is equivalent to ker(T) = {0}.

In our discussion of groups we remarked that one of the fundamental goals of the theory of finite groups is to make a list of all nonisomorphic finite groups, this goal being beyond the present state of knowledge. In the case of vector spaces, however, the situation is far different. The following theorem gives a complete classification of all finite-dimensional vector spaces over a field F

Theorem 4: Let V be a finite-dimensional vector space over a field F and let n = dimFV. Then Visomorphic withFn.

Proof: Let {e1,...,en} be a basis of V. Then every vector v element of V can be expressed uniquely in the form

v = alpha1e1 + ... + alphanen.

Define the function T:mapsFn by

T(alpha1e1 + ... + alphanen) = (alpha1,...,alphan).

It is easy to check that T is a linear transformation and T is clearly surjective. If v element of ker(T), then

T(v) = 0 implies(alpha1,...,alphan) = (0,...,0)
implies alpha1 = alpha2 = ... = alphan = 0
implies v = 0.

Thus, by Proposition 3, T is an isomorphism. Therefore, Visomorphic withFn.

Theorem 5: Let V and W be vector spaces over F, and let T:VmapsW be a linear transformation.

(1) dimF(T(V)) < dimF(V)

(2) If T is an isomorphism, dimF(T(V)) = dimF(V).

Proof: (1) Without loss of generality, let us assume that V is finite-dimensional, and let {e1,...,en} be a basis of V. Then

V = {alpha1e1,...,alphanen | ai element of F},

so that

T(V) = {alpha1T(e1) + ... + alphanT(en) | alphai element of F},

and thus {T(e1),...,T(en)} is a set of generators for T(V). In particular, dimF(T(V)) < n.

(2) Let the notation be as in the proof of (1). It suffices to show that {T(e1),...,T(en)} is a basis for T(V). Since we have already shown that this set generates T(V), it suffices to show that {T(e1),...,T(en)} is linearly independent. If alpha1,...,alphan element of F are such that

0 = alpha1T(e1) + ... + alphanT(en).

0 = T(alpha1e1 + ... + alphanen)

implies 0 = alpha1e1 + ... + alphanen   (since T is an isomorphism)

implies alpha1 = ... = alphan = 0   ({e1,...,en} is linearly independent)

implies {T(e1),...,T(en)} is linearly independent.

Theorem 5 has an interesting application to the theory of simultaneous linear equations. Let us consider the following system of equations:

a11x1 + a12x2 + ... + a1mxm = b1
a21x1 + a22x2 + ... + a2mxm = b2
.
(7)
.
.
an1x1 + an2x2 + ... + anmxm = bn,

where aij, bi all belong to a given field F. A solution of the system (7) is an m-tuple (x1,...,xm) element of Fm for which Equations (7) hold. A given system may or may not have a solution. In what follows we will study homogeneous systems - that is, systems of the form (7), in which b1 = b2 = ... = bn = 0. Such a system always has at least one solution, the zero solution (x1,...,xm) = (0,...,0). The following result guarantees the existence of nonzero solutions - that is solutions for which at least one xi is nonzero.

Theorem 6: Let

a11x1 + a12x2 + ... + a1mxm = 0
a21x1 + a22x2 + ... + a2mxm = 0
.
(8)
.
.
an1x1 + an2x2 + ... + anmxm = 0,

be a homogeneous system with coefficients aij belonging to the field F. If n < m, then the system always has a nonzero solution.

Proof: Consider the linear transformation T:FmmapsFn defined by T((x1,...,xm)) = (a11x1 + ... + a1mxm,...,an1x1 + ...+ anmxm element of Fn. Note that (x1,...,xm) is a solution of the system (8) is equivalent to

T((x1,...,xm)) = (0,...,0)
is equivalent to(x1,...,xm) element of ker(T).

Assume that (8) has only the zero solution. Then ker(T) = {0}. Therefore, T is an isomorphism. By Theorem 5,

(9)
dimF(T(Fm)) = dimF(Fm) = m.

However, since T(Fm) subset of Fn, we see that dimF(T(Fm)) < n. Therefore, by (9), m < n, which contradicts the assumption that n < m. Thus, (8) has nonzero solutions.
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