Linear Algebraic Aspects

Chungen Liu
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Chungen Liu: Guangzhou University, School of Mathematical and Information Science

Chapter Chapter 1 in Index theory in nonlinear analysis, 2019, pp 1-21 from Springer

Abstract: Abstract In this book, we define by ℕ , ℤ , ℝ $$\mathbb {N},\;\mathbb {Z},\;\mathbb {R}$$ and ℂ $${\mathbb {C}}$$ 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 ∈ ℕ $$n\in \mathbb {N}$$ 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 ℒ ( K n ) $$\mathbb L(K^n)$$ ℒ ( K n ) $$\mathbb L(K^n)$$ denote the group of all n × n matrices with entries in the field K, and ℒ s ( K n ) $$\mathbb L_s(K^n)$$ ℒ s ( K n ) $$\mathbb L_s(K^n)$$ the subset of ℒ ( K n ) $$\mathbb L(K^n)$$ consists of symmetric matrices. Any linear map T : K n → K n corresponds to a matrix T ∈ ℒ ( K n ) $$T\in \mathbb L (K^n)$$ in the usual way. We will not distinguish these two objectors.

Date: 2019
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DOI: 10.1007/978-981-13-7287-2_1

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