It's safe to do abdominal exercises lying on your back during your first trimester. Toward the end of the first trimester, or when you start "showing," you should stop doing abdominal exercises on your back (see Abdominal Exercises for other exercises you can do at this point).
When you are doing abdominal exercises, it's important to breathe correctly: remember to inhale to start and exhale on each effort.
The purpose of abdominal exercises is to strengthen core muscles. The deeper transverse abdominis muscle that runs horizontally across your body is vital for core stability and strength as your baby develops. The rectus abdominis muscle that runs vertically down your body is the muscle that will stretch during pregnancy and weaken, so it's vital to keep the transverse muscle strong to help your posture and support your spine.
The sooner you begin to strengthen the transverse abdominis, the better. In the first trimester, one of the exercises you can do to strengthen this muscle is shown below.