Question

The collection Mn of n-by-n matrices is a vector space in the usual way: You can add matrices component-wise and multiply all their entries by scalars. Really, it is just a repackaging of R n 2 . Consider the linear map S : Mn → Mn given by S(A) = (A−At )/2, where t is the transpose. (a) (2 points) Apply S to a 2-by-2 matrix of your choice (don’t pick a trivial matrix, like the identity or the zero matrix).

Answer 1

With
`A = \left[\begin{array}{cc}1&2\\4&5\end{array}\right]` , we have
`S(A) = \left[\begin{array}{cc}0&-1\\1&0\end{array}\right]`

Explanation:

Ok, lets apply the transformation S to the 2 by 2 matrix

`A = \left[\begin{array}{cc}1&2\\4&5\end{array}\right]`

The traspose of A is

`A^t = \left[\begin{array}{cc}1&4\\2&5\end{array}\right]`

Thus,

`S(A) = (A - A^t)/ 2 = A/2 - A^t/2 = \left[\begin{array}{cc}1/2&2/2\\4/2&5/2\end{array}\right] - \left[\begin{array}{cc}1/2&4/2\\2/2&5/2\end{array}\right] = \\\left[\begin{array}{cc}0&-1\\1&0\end{array}\right]`