Math · 05
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Do the vectors \((1,2)\) and \((2,4)\) form a basis of \(\mathbb{R}^2\)?
Timer not started. Twenty minutes of real work before a solution.
No. The second is twice the first, so they are linearly dependent and their span is a line, not the plane. A basis of \(\mathbb{R}^2\) needs two independent vectors.
Attack 2
Solve \(x+y=1\), \(2x+2y=2\), and describe the solution set.
Timer not started. Twenty minutes of real work before a solution.
The second equation is twice the first, so there is one constraint. The solutions are all \((x,1-x)\) for \(x\in\mathbb{R}\), a line.
Attack 3
Compute \(\det\begin{pmatrix}1&2\\3&4\end{pmatrix}\). What does the sign tell you?
Timer not started. Twenty minutes of real work before a solution.
\(1\cdot 4-2\cdot 3=-2\). The absolute value is the area of the parallelogram spanned by the columns. The negative sign means the ordered basis \((1,3)\), \((2,4)\) is opposite the standard orientation.
Linear algebra is the study of maps that send lines through the origin to lines through the origin, and of the finite-dimensional spaces those maps act on. Once dimension is finite, every abstract question becomes a question about matrices, and every matrix question is secretly about a linear map that did not want coordinates. The objects you must be able to move between without thinking are: span, independence, basis, dimension, rank, nullity, determinant, eigenvalues, inner products, and the singular value decomposition.
Strang’s MIT course is the computational and geometric introduction. Axler’s Linear Algebra Done Right, 4th edition, is free from the author and refuses to lead with determinants; that is the right second pass, not the first, because you still need to compute. Treil’s Linear Algebra Done Wrong is the free book that keeps determinants and still proves things. Learn Anything’s linear algebra page puts the 3Blue1Brown series and the Georgia Tech interactive text near the top. Both are used below. The rest of that page is software libraries and unrelated algebra, and it was ignored.
The Putnam problems that are “just linear algebra” are short and brutal. They assume you can see a matrix as a map.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Bases and dimension. A basis is a linearly independent spanning set. Every basis of a space has the same number of vectors; that number is the dimension. Any independent set extends to a basis, and any spanning set contains a basis.
The replacement theorem: if \(n\) vectors span and \(m\) vectors are independent, then \(m\le n\). Apply it both ways to two bases. Extension is replacement run until the span is the whole space.
Rank–nullity. For a linear map \(T:V\to W\) between finite-dimensional spaces, \(\dim\ker T+\dim\operatorname{im} T=\dim V\).
Take a basis of the kernel and extend it to a basis of \(V\). The images of the extra basis vectors are a basis of the image. Count.
Determinant. The determinant is the unique alternating multilinear form on columns that is 1 on the identity. It is zero exactly when the columns are dependent. It multiplies under matrix multiplication.
Existence is the Leibniz formula. Uniqueness is expansion along the definition by column operations. Multiplicativity can be seen by fixing one matrix and noticing both sides, as functions of the other, are alternating and multilinear and agree on the identity after scaling.
Eigenvalues. \(\lambda\) is an eigenvalue of \(T\) when \(Tv=\lambda v\) for some \(v\ne 0\). It is a root of the characteristic polynomial \(\det(A-\lambda I)=0\). A matrix is diagonalizable when there is a basis of eigenvectors, which happens in particular when the characteristic polynomial splits into distinct linear factors.
\(A-\lambda I\) has a nontrivial kernel exactly when its determinant vanishes. Distinct eigenvalues produce independent eigenvectors: a shortest dependence relation can be hit with \(A\) and subtracted to get a shorter one.
Inner products and SVD. An inner product induces a norm and orthogonality. An orthonormal set is independent. The orthogonal projection onto a subspace \(U\) is the unique vector in \(U\) whose error is orthogonal to \(U\). Every real \(m\times n\) matrix \(A\) factors as \(A=U\Sigma V^{T}\) with \(U,V\) orthogonal and \(\Sigma\) rectangular diagonal; the diagonal entries are the singular values, square roots of eigenvalues of \(A^{T}A\).
Projection: expand in an orthonormal basis of \(U\) and match coefficients, or minimize the quadratic. SVD: \(A^{T}A\) is symmetric positive semidefinite, hence orthogonally diagonalizable, and the nonzero eigenvalues give the squared singular values. The rest is bookkeeping with orthonormal bases of the domain and codomain.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
3Blue1Brown, Essence of Linear Algebra
Watch once, do not study from this. It is also the one Learn Anything linear-algebra link that earns the preview slot. Close it before you touch a problem set.
The thing you work through.
MIT 18.06 (Strang) and the Georgia Tech interactive text
18.06 is the course: lectures, problem sets, exams. The free book you can read beside it is Margalit and Rabinoff, Interactive Linear Algebra: https://textbooks.math.gatech.edu/ila/ . Strang, Introduction to Linear Algebra, ISBN 978-1-7331466-7-8, is the paid book the course tracks; the MIT page is https://math.mit.edu/~gs/linearalgebra/ . The free pair is better as a daily diet because the exams are public.
The problem book.
Axler, Linear Algebra Done Right, 4th edition, and Treil, Linear Algebra Done Wrong
Axler’s PDF is free from the author; ISBN 978-3-031-41025-3. Treil is the determinant-friendly proof book: https://www.math.brown.edu/streil/papers/LADW/LADW.html . Use Axler for the spectral theorem and Treil when you still need to compute a determinant without shame.
The hardest questions that exist at this level.
Borisov’s topic-sorted Putnam list, plus Matoušek’s miniatures
The PDF sorts Putnam problems by subject, including linear algebra. Matoušek, Thirty-three Miniatures, is a free PDF of serious applications: https://kam.mff.cuni.cz/~matousek/stml-53-matousek-1.pdf . Prasolov’s problem book was not verified as a stable free file, so it is not linked. Gelfand’s Lectures on Linear Algebra is the right Soviet companion and is paid; no free legal copy was verified.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: one MIT 18.06 exam, closed book, written in full, and five problems from Borisov’s linear-algebra section with complete proofs. You must also state and prove rank–nullity and the spectral theorem for a real symmetric matrix, from a blank page. Fail if singular value decomposition is a slogan you cannot connect to eigenvalues of \(A^{T}A\).
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Define a basis, prove rank–nullity, and explain what an eigenvalue is in terms of a line the map preserves. No notes.