Can you attach a lumber post base to a brick patio, just as like as to a concrete one?
Is an additional concrete footer needed?
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It depends on how much horizontal strength you need in the post. If you want it to hold up a fence vertical, then no. The sideways-leaning-tension would likely crack the bricks or mortar.
However, if you only need to support vertical weight then it would likely work. For example, one leg of a patio roof covering that is otherwise held rigid by attaching to the house. (In this case the house is providing the sideways strength for the roof, and the post is mainly providing vertical compression support.)
I would use a bracket like this:
It's a good idea to have a separation of the bottom of the wood from the brick or concrete, so moisture doesn't rot the post.
No. You won't get enough support. The bricks are just laying on the ground, held together by mortar. You need to cut out some mortar joints to open up room to dig a hole for the post. When done, re-mortar the bricks (you'll have to cut some to fit) back around the post.