43. Understanding Linear Algebra 3.5: Column Space, Null Space, and Rank

In this video, we learn how to find bases and dimensions for two important subspaces associated with a matrix: the column space Col(A) and the null space Nul(A). We introduce the concept of rank, work through several examples involving singular, invertible, and non-square matrices, and develop methods for finding bases for both spaces using row reduction. Along the way, we see why the pivot columns of the original matrix form a basis for the column space, derive a formula for the dimension of the null space, and conclude with the Rank-Nullity Theorem: dim Col(A) + dim Nul(A) = n Based on Section 3.5 of Understanding Linear Algebra by David Austin.