Definition and examples
q,(X) ~ q,'(X) in the vector space ('A) -4 cf>'('A) is a linear transforma- tion in the vector space ffi of polynomials of degree ('A) -4 cf>('A + 1) - cf>('A) is a linear trans- formation in ['A]
This section explores the combinations of linear transformations, noting that the findings are also applicable to the broader context of o-homomorphisms of modules However, for clarity and simplicity, we will focus on the specific case that is most relevant to our discussion.
Suppose first that A and B are linear transformations of a vec- tor space ffi l into the same space ffi2 We define a mapping
A + B of ffi l into ffi2 by the equation
The equation (3) x(A + B) = xA + xB holds true for any x in ffil, demonstrating that the transformation A + B affects x by combining the images xA and xB Consequently, A + B serves as a single-valued transformation from ffil to ffi2.
= xA + xB + yA + yB = x(A + B) + y(A + B) and
(ax) (A + B) = (ax)A + (ax)B = a(xA) + a(xB)
A + B is a linear transformation of ffi1 into ffi2
The set of all linear transformations from ffi l to ffi2, denoted as ~(ffih ffi2), forms a commutative group when combined with the addition and composition operations We can confirm that both the associative and commutative laws are satisfied within this framework.
34 LINEAR TRANSFORMATIONS x[(A + B) + C] = x(A + B) + xC = xA + xB + xC, x[A + (B + C)] = xA + x(B + C) = xA + xB + xC, x(A + B) = xA + xB, x(B + A) = xB + xA
In the context of vector transformations, both (A + B) + C and A + (B + C) yield the same result for any x in m h, illustrating their equality Additionally, the commutative property is demonstrated as A + B equals B + A We introduce the zero mapping, denoted as 0, defined by the condition xO = 0, where 0 represents the zero vector in m 2 This mapping confirms that A + 0 = A and 0 + A = A for all A in ~(mh m 2), establishing 0 as the identity element for additive composition Furthermore, for any A in ~(mh m 2), we define -A as the mapping where x(-A) = -xA It is straightforward to verify that -A belongs to ~(mh m 2) and serves as the inverse of A, since x(A + (-A)) equals xA - xA = 0 for all x.
This completes the verification that ~(mh m 2 ), + is a commuta- tive group
We introduce next a second composition for linear transforma- tions This is defined for any A in ~(mh m 2 ) and any B in
~(m2' m3), and it is taken to be the resultant of A followed by B
As usual, we denote the resultant as AB Hence by definition x(AB) = (xd)B Consequently
(x + y)(AB) = ôx + y)A)B = (xA + yA)B = (xA)B + (yA)B
(ax)(AB) = ôax)A)B = (a(xdằB = aôxd)B) = a(x(ABằ
This shows that AB e ~(mh m3)'
As is well known, the product AB is an associative one, that is, if A e ~(mh m2), B e ~(m2' ma) and C e ~(ma, m4), then
(AB)C = A(BC); xôAB)C) = (x(ABằC = ôxd)B)C x(A(BCằ = (xd)(BC) = ôxA)B)C
We prove next the important distributive laws: If A d~(~h ~2)'
These follow from the following equations: x(A(B + C)) = (xA)(B + C) = (xA)B + (xA)C
= x(AB) + x(AC) = x(AB + AC) x((B + C)D) = (xB + xC)D = (xB)D + (xC)D
We now specialize the foregoing results to the case of the linear transformations in a single vector space~ It is clear that
In the mathematical structure defined as a ring, 2(~, ~) with the operation + forms a commutative group The set 2(~, ~) is closed under the operation, which is both associative and distributive concerning addition Additionally, the identity mapping x ~ x exists within this structure, denoted as 1, serving as the identity element in the ring 2, satisfying the condition Al = A = lA for all elements A.
Suppose next that