Linear Combinations: Vectors and Matrices

This lecture introduces the concept of linear combinations of vectors and matrices, exploring how different vectors can be combined to generate new vectors in Rn. The instructor explains the operations of addition, scalar multiplication, and subtraction on vectors, highlighting the properties and geometric interpretations of these operations. Through examples, the lecture demonstrates how linear combinations of vectors can form lines, planes, or grids in Rn, depending on the number of vectors involved. The lecture also covers the use of matrices to represent linear transformations and how matrix-vector multiplication can be interpreted as a combination of the columns of the matrix. The discussion extends to solving systems of linear equations using matrix operations and understanding the span of vectors in Rn.