The Louisiana Purchase changed the way that the United States viewed itself because it caused the US to start to see itself as a potentially great power. Before the US bought Louisiana, it was a relatively small country hemmed in by other countries. After it made the purchase, it was on its way to becoming a country that stretched across the continent. This changed the way that Americans saw their country.
Before the Louisiana Purchase, the United States only stretched as far west as the Mississippi River. The lands that were to make up the Louisiana Purchase were controlled by Spain and were regained by France in 1800. During this time, the US was vulnerable to having its western trade closed off if Spain (or France) stopped allowing the US to use the port of New Orleans. The US was also unable to expand any further because Louisiana was in foreign hands. These two facts meant that the US, before 1803, could not really dream of dominating North America and could not even be sure that it would be able to continue to send goods down the Mississippi to be exported from New Orleans. This meant that America could not see itself as a secure country that could expect to become a power.
After the Louisiana Purchase, all this changed. America could now feel that it was free to grow as much as it wanted to. Americans could feel like they were in control of their own destiny. This gave them a greater feeling of independence and a feeling that they could become a world power. Thus, we can say that the Louisiana Purchase changed the way the US viewed itself by allowing the US to believe that it would become a world power.