Jordan canonical form and abstract dual spaces are excluded.
Further resources: Gilbert Strang’s "Introduction to Linear Algebra" (MIT), 3Blue1Brown’s "Essence of Linear Algebra" video series (YouTube), and practical coding with NumPy’s linear algebra module. Introduction To Linear Algebra For Science And Engineering
The most powerful concept. An expression ( c_1\vecv_1 + c_2\vecv_2 + \dots + c_n\vecv_n ) forms a linear combination. In signal processing, any sound wave can be represented as a linear combination of sine waves (Fourier series). In chemistry, a molecule’s properties are linear combinations of atomic orbitals. Jordan canonical form and abstract dual spaces are excluded
Eigenvectors are the hidden skeleton of a linear transformation. An eigenvector ( \vecv ) of ( A ) satisfies ( A\vecv = \lambda\vecv ), meaning the vector only gets scaled (by ( \lambda ), the eigenvalue), not rotated. An expression ( c_1\vecv_1 + c_2\vecv_2 + \dots
At its core, linear algebra is the study of linear equations and their representations in multi-dimensional space. In science and engineering, we rarely deal with a single variable. Instead, we observe systems where dozens or even millions of variables interact simultaneously. Linear algebra allows us to organize these variables into structures called vectors and matrices, making it possible to perform massive calculations efficiently.