Final answer:
The Danube River is the river that flows through Linz and Vienna in Austria, playing a crucial role in transportation, the economy, and cultural life.
Step-by-step explanation:
The river that runs through the cities of Linz and Vienna in Austria is the Danube River. This iconic river is essential for transport, trade, and the overall cultural and historical identity of Central Europe. In Austria, the Danube enhances the picturesque landscapes and supports the high standard of living by facilitating economic activity. Cities such as Vienna, which has a rich musical and artistic heritage, and Linz, which is becoming a cultural European center, are among the many urban centers the river traverses.
Apart from its significance in modern times, rivers like the Danube have historically provided vital transportation routes throughout Europe, contributing to the development of its cities and trade systems. The Danube, in particular, flows east through various major European cities before emptying into the Black Sea, marking its importance not just in Austria but across the continent.