Explore the fundamental concepts of simplices and simplicial complexes in this comprehensive lecture on Algebraic Topology. Delve into convex combinations and convex hulls, examining the natural progression from points to line segments, triangles, and tetrahedra. Learn about the standard representation of simplices as convex hulls of unit basis vectors in vector spaces. Discover how simplicial complexes are formed and understand the importance of orientations in simplices. Investigate the relationship between a tetrahedron's orientation and its triangular faces, and grasp the crucial concept of boundary operations in simplicial complexes. Gain insights into the foundation of homology theory through the exploration of these essential algebraic topology concepts.