Noah completely trusted God

Noah put all his trust in the Lord

The author emphasizes his point more by beginning each sentence with "by faith": By faith Noah, when warned about things not yet seen, in holy fear built an ark to save his family. By his faith he condemned the world and became heir of the righteousness that is in keeping with faith (v. 7). For Noah, the author has more biblical information: God warned Noah about a flood, told him to build an ark, and Noah obeyed and saved himself and his family (Genesis 6-9).
Noah didn’t really condemn the world (God did that, based on their behavior), but his faithful example made it obvious how hopelessly evil the world had become—no one repented even after 120 years of warning. And by his faith Noah became an heir of righteousness—he is the first person in the Bible to be called righteous (Gen. 7:1). As it would later be said for Abraham, "His faith was counted as righteousness" (Gen. 15:6). He was considered right with God because he was faithful.
The Greek word pistis can mean either faith or faithfulness, and many people have noticed that Hebrews often uses it in the sense of faithfulness, or obedience, and it is sometimes hard to tell whether the author is focusing on belief or behavior. (Although Paul occasionally uses the same word in the sense of faithfulness, he usually refers to belief.) Obedience is evidence of belief, and both are needed. Noah did what God told him to do because he trusted God—he believed that God would condemn the wicked and save Noah and his family if they built an ark.