Linear Algebra

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...