Articles

From Dot Product to Inner Product Space: The Unified Language Behind Linear Algebra, Signals, and AI Abstract The **inner product** is a core algebraic structure shared across linear algebra, functional analysis, signal processing, machine learning, and quantum mechanics. Taking the inner product as its sole thread, this paper begins with the dot product in finite-dimensional Euclidean space and progressively introduces the axioms of inner product spaces, orthogonal decomposition, least-squares projection, Hilbert spaces, Fourier series and transforms, convolution, the discrete cosine transform, wavelet analysis, self-attention mechanisms, kernel methods, and state-vector projection in quantum mechanics. It reveals the mathematical unity underlying these seemingly disparate concep...

Introduction This article systematically elaborates on the impedance characteristics and physical significance of resistors, capacitors, and inductors in alternating current (AC) circuits. Through phasor methods and frequency-domain analysis, the impedance formulas for these three components are derived, revealing the phase relationships between voltage and current as well as the patterns of energy variation. Furthermore, the mathematical expressions and frequency characteristics of resistance, capacitive reactance, and inductive reactance are compared and summarized. This facilitates an intuitive understanding of classic principles such as "passing AC while blocking DC" and "passing DC while blocking AC," providing a theoretical foundation for subsequent applications including filte...