Linear Algebra: Change of Basis and Matrix Representation

This lecture covers the concept of changing bases in vector spaces, focusing on the transition between two different bases in linear algebra. The instructor explains how to represent linear transformations using matrices, emphasizing the importance of understanding the relationship between bases and matrices. Key topics include the dimension of vector spaces, the image and kernel of matrices, and the theorem of rank. The lecture also delves into the space of rows and its connection to the image of a matrix. Through theoretical explanations and examples, students learn how to determine if two spans are equal and how to find the matrix representation of a linear transformation.