Learn about vectors and spinors in quantum mechanics through this 21-minute lecture from the nanoHUB-U course "Fundamentals of Nanoelectronics Part B: Quantum Transport." Explore the mathematical foundations essential for understanding quantum transport phenomena in nanoelectronic devices, building upon concepts in linear algebra to examine how vectors and spinors describe quantum states in atomic-scale systems. Discover the role these mathematical tools play in analyzing current flow at the nanoscale, where billions of nanotransistors with active regions spanning just hundreds of atoms enable modern smartphone technology. Gain insights into advanced concepts in non-equilibrium statistical mechanics while developing the mathematical framework necessary for understanding quantum transport in nanoelectronic applications.