Lecture Note on Linear Algebra 8. Matrix Operations

In the previous lectures, we have seen that matrices play an important role in solving system of linear equations and in studying linear transformations from Rn to Rm. Let Rmn denote the set of all m×n matrices over R. A column vector of dimension n is defined to be an n × 1 matrix. It turns out that Rn = Rn1. Recall that we have defined addition and scalar multiplication on Rn for any n> 1. We also defined the product of a matrix and a vector. In this note, these operations are extended to the operations between matrices. Basic concept：diagonal entries(对角线元素), main diagonal(主对角线), zero matrix(零矩阵), square matrix(方阵), diagonal matrix(对角线矩阵), identity matrix(单位矩阵), transpose(转置), matrix inverse(矩阵的逆), ele-mentary matrix (初等矩阵) Question: Why do we need matrix computation? 1. A linear transformation is corresponding to a unique matrix 2. For two linear transformations TA and TB, what is the combination of two linear transform, i.e. (TA + TB)(x⃗)? 3. What is compound transformation TA ◦ TB(x⃗)?