Linear Algebraic Aspects

Abstract

In this book, we define by $\mathbb {N},\;\mathbb {Z},\;\mathbb {R}$ and ${\mathbb {C}}$ the sets of all natural, integral, real and complex numbers respectively. For a matrix M, we denote its transpose by M ^T. For any $n\in \mathbb {N}$ and any field K, denote by K ⁿ the linear space formed by all the column vectors of the form x = (x ₁, ⋯ , x _n)^T with x _i ∈ K. We usually treat x ∈ K ⁿ as an n × 1 matrix with no explain. Let $\mathbb L(K^n)$ denote the group of all n × n matrices with entries in the field K, and $\mathbb L_s(K^n)$ the subset of $\mathbb L(K^n)$ consists of symmetric matrices. Any linear map T : K ⁿ → K ⁿ corresponds to a matrix $T\in \mathbb L (K^n)$ in the usual way. We will not distinguish these two objectors.