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3. Dual transformation

A linear transformation T: VW induces a transformation

T*(f) = fT: W*V*

between the dual spaces, called the dual transformation. Specifically, the formula means that T*(f) is a linear function on V, and the value of the function at vV is T*(f)(v) = f(T(v)). The generalization of T* to Hom spaces is given in this exercise.

The following verifies that T*(f + g) = T*(f) + T*(g).

T*(f + g)(v)
= (f + g)(T(v)) (definition of T*)
= f(T(v)) + g(T(v)) (definition of the addition f + g)
= T*(f)(v) + T*(g)(v) (definition of T*)
= (T*(f) + T*(g))(v). (definition of the addition T*(f) + T*(g))

The equality T*(cf) = cT*(f) can be similarly verified. Thus the dual transformation is linear.

Next we compute the dual transformation of a linear transformation T(x) = Ax: RnRm given by an m by n matrix A. By making use of the identification ι: (Rn)*Rn, the dual transformation T*: (Rm)* → (Rn)* may be identified with a linear transformation T' = ιT*ι-1: RmRn. The matrix of this transformation (the red question in the picture) may be regarded as the matrix of the dual transformation.

To find the matrix for T', we try the case that

A = [ a11 a12 ], or T(x1, x2) = (a11x1 + a12x2, a21x1 + a22x2, a31x1 + a32x2): R2R3.
a21 a22
a31 a32

For the inear function f(y1, y2, y3) = c1y1 + c2y2 + c3y3 ∈ (R3)*, we have

T*(f)(x1, x2) = f(T(x1, x2))
= f(a11x1 + a12x2, a21x1 + a22x2, a31x1 + a32x2)
= c1(a11x1 + a12x2) + c2(a21x1 + a22x2) + c3(a31x1 + a32x2)
= (a11c1 + a21c2 + a31c3)x1 + (a12c1 + a22c2 + a32c3)x2 ∈ (R2)*.

The transformation T' takes ι(f) = (c1, c2, c3) to ι(T*(f)) = (a11c1 + a21c2 + a31c3, a12c1 + a22c2 + a32c3):

T'[ c1 ] =
[ a11c1 + a21c2 + a31c3 ] = [ a11 a21 a31 ]
a11c1 + a21c2 + a31c3 a12 a22 a32
[ c1 ] = AT[ c1 ].
c2 c2 c2
c3 c3 c3

Thus the matrix for T' is AT. In general, we have the following.

Under the identification ι: (Rn)*Rn, if a linear transformation is given by a matrix A, then the dual transformation is given by the matrix AT.

The fact tells us that the dual transformation is, up to the identification ι, a generalization of the transpose. The linearity and product properties of the transpose then follow from the following properties of the dual. The explanation for the involution property is much more complicated and appears here.

linearity
(T + S)* = T* + S*, (cT)* = cT*
product
(TS)* = S*T*

Similar properties can be proved for the generalization to the Hom space.

Proof We prove the the product property. Let S: UV and T: VW be linear transformations. Then for fW*, and uU, we have

(TS)*(f)(u)
= f(TS(u)) (definition of dual (TS)*)
= f(T(S(u))) (definition of composition TS)
= T*(f)(S(u)) (definition of dual T*)
= S*(T*(f))(u) (definition of dual S*)
= (S*T*)(f)(u). (definition of composition S*T*)

Thus (TS)*(f)(u) = (S*T*)(f)(u) for any f and u. This means (TS)* = S*T*.

The proof of linearity is similar.


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