Linear Algebra

What this covers

• Gaussian elimination and reduced row echelon form, including reading free variables and parametric solution sets off the RREF
• Span, linear independence, and how to test them by setting up and solving the right homogeneous system
• Subspaces, bases, and dimension, including finding bases for column space, row space, and null space, and applying the rank-nullity theorem
• Linear transformations and their standard matrices; kernel, range, injectivity, and surjectivity in matrix terms
• Determinants via cofactor expansion and row operations, and the invertible matrix theorem as a list of equivalent conditions
• Eigenvalues, eigenvectors, characteristic polynomials, and diagonalization, including when a matrix fails to diagonalize
• Orthogonality, Gram-Schmidt, orthogonal projection, and least-squares solutions to inconsistent systems

Where learners get stuck

Treating linear independence as a property of individual vectors rather than of a set

Students say 'this vector is independent' because the definition is usually first met as a computation (row reduce, check for a pivot in every column) rather than as a statement about which combinations equal zero. The habit breaks down when they need to argue that adding a vector to a spanning set destroys independence.

Confusing the column space with the null space, or looking for a basis of the column space in the RREF

Both are found from the same row reduction, so the procedures blur together. Row operations preserve the null space and column dependencies but change the column space itself, so basis vectors must come from the original matrix's pivot columns, not the reduced one.

Assuming any matrix with repeated eigenvalues cannot be diagonalized, or that any n×n matrix with n distinct eigenvalues is the only diagonalizable case

The distinct-eigenvalues theorem is a sufficient condition, not a necessary one, and it gets memorised as a rule. Diagonalizability actually depends on geometric multiplicity matching algebraic multiplicity, which requires computing the eigenspace dimension, not just the characteristic polynomial.

Believing det(A + B) = det(A) + det(B) because determinants feel linear

The determinant is linear in each row separately, and textbooks state this multilinearity early. Students generalise it to the whole matrix, which also makes the multiplicative property det(AB) = det(A)det(B) look arbitrary rather than notable.

What a session looks like

A session usually starts with you stating a problem or theorem you're stuck on and describing your current approach. Evelyn asks what the definition actually requires before any arithmetic starts — for instance, what equation you need to solve to test independence — then works through the computation with you, checking intermediate row operations and asking you to interpret each result rather than just report it. For proof-based questions, you're asked to say the argument aloud and defend the step that assumes too much. Sessions typically end with you restating one definition or theorem in your own words.

Helpful to know first

• Comfort with algebraic manipulation: solving systems of two or three equations, factoring polynomials, and working with variables symbolically
• Familiarity with function notation and the idea of domain and range
• Basic 2D and 3D coordinate geometry, including vectors as arrows and the dot product (helpful but often re-covered)
• No calculus is required for most first courses, though some texts use it in applications

Questions

Is linear algebra harder than Calculus II?

It is harder in a different way. The computations are mostly arithmetic rather than technique-heavy integration, but the course asks for proofs and abstract definitions earlier than calculus does. Students who did well in calculus by pattern-matching problem types often struggle with the first proof-based assignment on subspaces.

Do I need to have taken calculus before linear algebra?

Most introductory linear algebra courses do not use calculus, though many schools list Calculus I as a prerequisite for mathematical maturity rather than content. Check your syllabus; if it covers differential equations applications or function spaces, some calculus background will be assumed.

Can Evelyn help with proof-writing, not just computation?

Yes. Sessions can focus on structuring proofs about subspaces, independence, or invertibility — identifying what you're allowed to assume, what needs to be shown, and where an argument only proves one direction of an 'if and only if'.

What's the fastest way to get better at eigenvalue problems?

Usually by separating the three steps that get merged: computing the characteristic polynomial, finding each eigenspace as a null space, and checking whether the eigenvectors form a basis. Most errors trace back to stopping after the eigenvalues without actually solving (A − λI)x = 0 properly.