When Paul says, "know you therefore," he is instructing them to learn from this case (Psalm 100:3; Luke 21:31; Hebrews 13:23).
The Greek word translated "of" here is "a primary preposition denoting origin (the point whence action or motion proceeds)." In this case, it is referring to those people whose origin or source is faith—those who have the right faith, which have been justified by God. This is the starting point of their spiritual life. Paul is showing that we are children of Abraham based on what we believe (have faith in), which is then evidenced by the way we live our life.
One of the many disputes that Christ had with the Jews during His time was over lineage. The Jews traced their physical lineage back to Abraham, and thus considered themselves to be children of Abraham. Christ disagreed with this because, if they were Abraham's sons in the fullest sense, they would have believed and acted just as Abraham did. The real children of Abraham are not his natural descendants (Matthew 3:9), but those who share his faith. This is why Abraham is called the "father of the faithful" (see also Luke 3:8; Romans 2:28-29, 9:6; James 2:21-23).
It is interesting to note again the link between faith (belief) and works (action). In this verse, Paul says that the true children of Abraham are the ones who have the same faith that Abraham had. Yet in John 8:39-41, Christ says the Jews really are not Abraham's children because their works—their actions—were not in accordance with what Abraham had done. There certainly is no contradiction here; the faith that made Abraham remarkable is the faith that motivates people to do good works (James 2:17-26)!
David C. Grabbe
Here, the subject is the role faith and works play in the justification process. This time, he uses Abraham as the model by which all his "children in the faith" also become "children of God." He begins by posing a question, which can be paraphrased as, "Do miracles come by ritual?" There is in this a veiled illusion to magic. Do miracles come by incantation? Do they come by knowing certain formulas that may include even such things as cutting the flesh or going through long periods of fasting or sufferings to get God's attention? Will God respond with a miracle out of pity once we show Him how humble and righteous we are? No, it does not work that way. Miracles come by a living God, who is actively working in our lives because He called us and we have faith in Him.
With that foundation, Paul begins what turns into the preamble for a very controversial section of Galatians. He proceeds to state that it was through faith that Abraham was justified. It is good to remember that Abraham not only believed who God is, but he also believed what God said. This is what set him apart from everybody else. His faith was not merely an intellectual agreement, but he also lived His faith.
Abraham's works did not win him acceptance by God, but they did prove to God that Abraham really did believe Him. So Paul says in verses 10-11 that those who rely on their works to justify them are under the curse of the law. What is "the curse of the law"? The death penalty! When one sins, he brings on himself the curse of the law he broke, which is death. In effect, he says that those who seek justifcation through works are still under the curse because justification by this means is impossible.
So powerful is the curse of the law that, when our sins were laid on the sinless Jesus Christ, the law claimed its due. Jesus died! Paul quotes from Deuteronomy 27:26 to counteract those who were troubling the church, because they were saying that their asceticism, magic, and similar things (like keeping Halakah, the oral law and traditions of Judaism) could justify.
John W. Ritenbaugh
The Covenants, Grace, and Law (Part 26)
Other Forerunner Commentary entries containing Galatians 3:7: