Question

show that linear operator ba need not be the same as ab in real analysis

Answer 1

A linear operator can be written as a matrix. It's easy to give an example of two matrices which do not commute.

For example,

$\left[\begin{array}{cc}1&2\\4&5\\\end{array}\right] \left[\begin{array}{cc}1&2\\1&4\\\end{array}\right] = \left[\begin{array}{cc}3&10\\9&28\\\end{array}\right]$

but

$\left[\begin{array}{cc}1&2\\1&4\\\end{array}\right]\left[\begin{array}{cc}1&2\\4&5\\\end{array}\right] = \left[\begin{array}{cc}9&12\\17&22\\\end{array}\right].$