find adjoint of a linear transformation defined by a cross product

Question

Suppose that a in R^3 is given. Define A in L(R^3) by Av = a (cross)v find adjoint A, assuming that R^3 is equipped with the standard dot product I know to find adjoint A, I must start from definition (Av, y) = (v, A* y) However, when replace Av = a x v, I don't know how to do more ((a x v), y)= (a,y)x (v,y) Someone helps me, please

Answer 1

You can solve the problem without computations by geometrical reasoning. The triple product $$ \langle a\times v , y\rangle $$ equals the signed volume of the parallelogram spanned by the vectors $a, v, y$, in that order. To write it in adjoint form you need to fill in the blank in the following formula: $$ \langle a\times v , y\rangle=\langle v, ?\rangle .$$ If you set $?=y\times a$ then you get for the left hand side the triple product of $v, a, y$, that is, the signed volume of the same parallelogram as before BUT with two sides permuted. Therefore, the sign changes and the correct answer is $$ \langle a\times v , y\rangle=-\langle v, a\times y\rangle .$$

(Of course you could have arrived at this result with a more algebraic approach, and probably if this is homework you'd better do so.)

Answer 2

Given a vector $a\in\mathbb R$, the matrix representation of the linear transformation $A:v\mapsto a\times v$ with respect to the standard basis, usually denoted by $[a]_\times$, is a skew-symmetric matrix known as the cross product matrix. Therefore the matrix of $A^\ast$ is $[a]_\times^T=-[a]_\times$, i.e. $A^\ast v=-av$.

Answer 3

computing Av = a x v let a =(a1,a2, a3) and v = (v1,v2,v3) apply cross product, I got Av = a2v3-a3v2-a1v3+a3v1+a1v2-a2v1 under the matrix form, it is Av = {{0, -a2, a3},{a1,0,-a3},{{-a1,a2,0}}{v1,v2,v3} so that A = [col1 col2 col3] as shown A = A^t