My solution attached.


NOTE that there is a typo in the statement of this exercise: The second-last line should actually read:



Currently it states that the dimension is 2j rather than 2j+1.

This is incorrect, because the main text says that m runs from -j to j, so it has 2j+1 possible values; and it is stated that has 2j+1 independent components. My solution also arrives at 2j+1 as the dimension of the representation space.