Proving Isomorphism of Skew-Symmetric Matrices to 3D Vectors

[Dollydaggerxo]

Homework Statement

Hello, I have been asked to prove that three different matrices which are skew-symmetric with a defined operation can be shown to be isomorphic to the usual vectors in 3d space with the operation of the cross product.

Homework Equations

Well the operation i guess is not so important to state as I have constructed a multiplication table for it but it is X*Y = XY-YX.

The Attempt at a Solution

I know that the cross product for a x b would be a2b3 - a3b2 etc etc and i can see in my head why it is isomorphic I just really do not know how to go about proving it. what should be vectors be, should they all be the same at just i,j,k? i don't know how to prove it without using numbers as the vectors.
please help

 

[Dollydaggerxo]

also how can i write a skew-symmetric matrix as a linear combination of I J and K?
please help thanks

 

[HallsofIvy]

If, by "I, J, and K" you mean the 3-d basis vectors, you can't- matrices are not vectors. But you can write such a matrix as a linear combination of three basis matrices that the isomorphism maps into $\vec{i}$, $\vec{j}$, and $\vec{k}$.

Any 3 by 3 skew-symmetric matrix is of the form
$$\begin{bmatrix}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{bmatrix}$$
$$= a\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0\end{bmatrix}+ c\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{bmatrix}$$

 

[Dollydaggerxo]

thanks so muchhh i think i know what I am doing now! can't believe never knew that way of writing skew-symmetric matrices

 

[Deveno]

actually, that operation IS important, it is the typical way of defining a Lie Bracket on a matrix algebra:

[X,Y] = XY - YX

by identifying R^3 with a certain subset of the matrix algebra, you have shown that the cross product can be used to identify R^3 as a subalgebra of the Lie Algebra M3(R).