Linear Algebra — Between the Lines
From the axioms of vector spaces to inner product spaces. A six-part series that answers the questions textbooks leave between the lines, building intuition for linear algebra.
Why define vector spaces axiomatically? From arrows to axioms
In high school, vectors are arrows. In a university course, they become elements of a set satisfying eight axioms. Why the abstraction? Because polynomials, matrices, and functions are all vector spaces — and the axioms are what let us treat them with a single theory.
What is "dimension," really? The truth about degrees of freedom
We all say "three-dimensional space" without blinking — but what exactly does the "three" mean? The answer is less obvious than it seems, and proving it requires the Steinitz exchange lemma.
Why matrix multiplication works that way: when linear maps become matrices
The definition of matrix multiplication looks arbitrary — until you realize it encodes composition of linear maps. We compute a concrete composition by hand and watch the "row-times-column" rule emerge naturally.
What does the determinant measure? Area, volume, and orientation
The textbook definition of the determinant — a sum over permutations — can feel like it dropped out of the sky. In fact, it measures something beautifully concrete: the signed volume scaling factor of a linear map.
Why eigenvalues matter: how diagonalization simplifies everything
Eigenvalues and eigenvectors appear suddenly in every linear algebra course — but why are they so important? Because they decompose a complicated linear map into independent scalings, turning hard problems into easy ones.
The geometry that inner products unlock: orthogonality, projection, and least squares
A vector space, by itself, has no concept of length or angle. Inner products supply both — and with them come orthogonal projections, the Gram–Schmidt process, least squares, and the bridge to Fourier analysis.