Linear Algebra

• Intuitive
• Concrete
• Abstract

Intuitive

Linear algebra gives you mini-spreadsheets for your math equations.

We can take a table of data (a matrix) and create updated tables from the original. It’s the power of a spreadsheet written as an equation. An Intuitive Guide to Linear Algebra by Kalid Azad

Concrete

• A great introduction is An Intuitive Guide to Linear Algebra by Kalid Azad
• For a quick introduction, see https://minireference.com/static/tutorials/linear_algebra_in_4_pages.pdf
• and http://www.cns.nyu.edu/~eero/NOTES/geomLinAlg.pdf which explains really nicely many linear algebra in geometric terms.
• For a nice guide, see How to Self-study Algebra: Linear Algebra
• Graphical Linear Algebra
• immersive linear algebra by J. Ström, K. Åström, and T. Akenine-Möller (free online book)
• Essence of Linear Algebra video mini-series

Good books are:

• Linear Algebra Done Right by Sheldon Axler - despite the title a great book.
• Linear Algebra Done Wrong by Treil - even better than Linear Algebra done Right.
• No bullshit guide to linear algebra by Ivan Savov
• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John H. Hubbard and Barbara Burke Hubbard - Extremely student friendly, lots of margin notes that talk about the "soft" stuff that's so crucial to the actual practice of math. Reading just the margins jumps your mathematical maturity by 2 years.

A nice video introduction is

• Essence of linear algebra by 3Blue1Brown

Abstract

Fix a field $K$ (in physics $\mathbb{R}$ or $\mathbb{C}$). A vector space over $K$ is an abelian group $V$ with a scalar multiplication satisfying the usual distributive and associative laws, and a linear map is a map $T: V \to W$ with $T(av + bw) = aT(v) + bT(w)$. Every vector space has a basis, and all bases have the same cardinality, the dimension. A choice of basis identifies an $n$-dimensional $V$ with $K^n$ and linear maps with matrices; the basis-independent content is what survives a change of basis $T \mapsto S^{-1} T S$.

Kernel, image and dimension. For $T: V \to W$ with $V$ finite-dimensional, the rank-nullity theorem states

$$\dim V = \dim \ker T + \dim \operatorname{im} T.$$

Duality and tensors. The dual space $V^* = \mathrm{Hom}(V, K)$ has the dual basis $e^i(e_j) = \delta^i_j$. A linear map $T$ induces the transpose $T^*: W^* \to V^*$, $T^*(\alpha) = \alpha \circ T$. The tensor product $V \otimes W$ is defined by its universal property: bilinear maps $V \times W \to U$ correspond to linear maps $V \otimes W \to U$. For finite-dimensional spaces $\mathrm{Hom}(V, W) \cong V^* \otimes W$, and a tensor of type $(r, s)$ is an element of $V^{\otimes r} \otimes (V^*)^{\otimes s}$. Upper and lower indices in physics notation record whether a slot belongs to $V$ or to $V^*$. A nondegenerate bilinear form $g$ gives an isomorphism $V \to V^*$, $v \mapsto g(v, \cdot)$, which is index lowering.

Determinant. The top exterior power $\Lambda^n V$ of an $n$-dimensional space is one-dimensional, so the induced map $\Lambda^n T$ is multiplication by a number. That number is $\det T$. This definition makes $\det(ST) = \det S \det T$ immediate and shows that $\det T$ is the factor by which $T$ scales oriented volumes.

Normal forms. Over an algebraically closed field such as $\mathbb{C}$, every operator on a finite-dimensional space has an eigenvalue, and the Jordan normal form classifies operators up to similarity. On a finite-dimensional complex inner product space, an operator is unitarily diagonalizable if and only if it is normal, $TT^\dagger = T^\dagger T$; self-adjoint operators have real eigenvalues and an orthonormal eigenbasis. This spectral theorem is the finite-dimensional model of observables in quantum mechanics; its infinite-dimensional version is formulated on a Hilbert space. Real symmetric bilinear forms are classified by Sylvester's law of inertia: there is a basis in which the form is $\mathrm{diag}(+1, \dots, +1, -1, \dots, -1, 0, \dots, 0)$, and the numbers of each sign are invariants. The Minkowski metric is the case with signature $(1, 3)$ or $(3, 1)$, depending on convention.

In the language of category theory, finite-dimensional vector spaces and linear maps form the category $\mathrm{Vect}_K$, and a representation of a group $G$ is a homomorphism $G \to GL(V)$. Linear algebra is the setting of representation theory, which is how symmetries act on quantum states.

Leads to: Covariant and Contravariant Coordinates, Dirac Notation, Exterior Product, Group Theory, Hilbert Space, Manifold, Product Spaces, Representation Theory, Tensor Calculus, Vector Calculus.