To what extent was British industrialization dependent on British imperialism? Please provide examples.

There was most certainly a very close relationship between industrialization and imperialism, and the two do tend to be very closely intertwined.
Ultimately, industrial economies are driven by the production and sale of mass-produced goods. In the example of British industrialization, this was initially focused largely in the textiles industry, and in this example you can see the connection between industrialization and empire: to run a textile industry, one needs large quantities of cloth. There is the example of the American colonies, where raw materials were sent back to the mother country to be converted into finished products. A similar story emerges in British India, where the domestic textile industry was essentially dismantled in favor of cash crop agriculture, with cotton being grown for export back to the British industrial centers and then sold back to India as finished clothes. This illustrates the fundamental economic relationship of empire.
However, just as empire was a key contributor to industrialization, the reverse applies as well. The needs of industrialized economies drove industrializing and industrial nations to seek out and compete over colonial holdings. Furthermore, Industrialization carried with it a dramatic increase in economic, political, and military power, which contributed to the dramatic expansion of European empire seen in the nineteenth century. Pre-industrial European powers were simply unable to impose themselves militarily the way that the industrial powers were able to, especially as science and technology continued to advance.