42. Understanding Linear Algebra 3.5: Subspaces, Bases, and Dimension

In this video, we introduce the concept of a subspace of R^p and connect it to two important spaces associated with a matrix: the span of its columns and the solution set of the homogeneous equation Ax = 0. Using an example, we explore how these sets form subspaces and discuss the ideas of basis and dimension for a subspace. We then define two fundamental subspaces in linear algebra: the column space Col(A) and the null space Nul(A). These spaces will play a central role throughout the remainder of the course. Based on Section 3.5 of Understanding Linear Algebra by David Austin.