U of T · Algebra I · Fall

MAT240H1: linear algebra, done with proofs.

Vector spaces over arbitrary fields, linear transformations, determinants, and eigenvalues — the proof-based linear algebra that pairs with the analysis stream and underpins everything in statistics, ML, and finance.

About this course

The abstract core tool.

MAT240H1 "Algebra I": vector spaces over arbitrary fields (including $\mathbb{C}$ and modular arithmetic), subspaces, bases and dimension, linear transformations, matrices, change of basis, similarity, determinants, polynomials over fields, eigenvalues and eigenvectors, characteristic polynomials, diagonalization, minimal polynomials, and the Cayley–Hamilton theorem. This guide uses Friedberg, Insel & Spence as a companion text. Co-requisite: MAT158H1 + MAT159H1. The sequel is MAT247H1 (Algebra II).

01

Chapters

In course order. Each page has definitions, theorems, worked proofs, pitfalls, and practice.

MAT240H1Algebra I · Fall MAT247H1Algebra II · Winter (sequel) Co-reqMAT158H1 + MAT159H1

  1. Fields & vector spaces

    Fields, the axioms, and first examples.

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  2. Subspaces, span & independence

    Linear combinations and linear independence.

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  3. Bases & dimension

    Existence of bases; the dimension theorem.

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  4. Linear transformations

    Maps, null space and range.

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  5. Rank–nullity & matrices

    $\dim V=\operatorname{rank}(T)+\operatorname{nullity}(T)$; matrix forms.

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  6. Change of basis & similarity

    Coordinate changes and similar matrices.

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  7. Systems & elementary operations

    Row reduction, rank, and inverses.

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  8. Determinants

    Definition, properties, $\det(AB)=\det A\det B$.

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  9. Polynomials, UFD & resultants

    Division, unique factorization, resultants, and minimal polynomials.

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  10. Eigenvalues & diagonalization

    Characteristic polynomial, diagonalizability, Cayley–Hamilton.

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02

Textbook Spine

The public calendar confirms MAT240H1's proof-based linear algebra topics. This guide uses Friedberg, Insel & Spence, Linear Algebra, as the companion text because its abstract-vector-space order matches the course: fields and vector spaces before matrices. Check the current syllabus for the exact edition and assigned problems.

Unit-to-textbook alignment
Vector spaces first

Units 1-3 pair with the text's opening chapters on fields, vector spaces, subspaces, span, independence, bases, and dimension. The practice target is proof from axioms, not row reduction.

Maps and matrices

Units 4-7 pair with linear transformations, rank-nullity, matrix representations, change of basis, similarity, systems, elementary operations, and inverses.

Structure theorems

Units 8-10 pair with determinants, polynomials over fields, eigenvalues, diagonalization, minimal polynomials, and Cayley-Hamilton. This is where computations become structural statements about operators.

DEFINE

Start from objects

Always name the field, vector space, basis, and linear map before writing a matrix. The same matrix can represent different maps under different bases.

PROVE

Use theorem templates

Rank-nullity, dimension, determinant multiplicativity, and diagonalization each have standard proof shapes. Memorize the shape, not the wording.

COMPUTE

Compute with meaning

Every row reduction, determinant, coordinate vector, or characteristic polynomial should answer a structural question: injective, spanning, invertible, similar, or diagonalizable?

COMPARE

Use multiple fields

Redo small examples over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Z}_p$. Many MAT240 pitfalls come from assuming real-number behavior over every field.

03

Practice Plan

MAT240 practice should alternate between proof and computation. The unit pages give guided questions; use this course-level mix to build exam readiness.

01

Axiom checks

For every new object, check closure, zero, inverses, and scalar compatibility before using vector-space theorems.

02

Basis discipline

Convert between coordinate-free statements and coordinate matrices. Write the basis explicitly every time.

03

Proof-computation pairs

Pair each computation with a proof question: row reduction with rank-nullity, determinants with invertibility, eigenvectors with invariant subspaces.

04

Counterexample sets

Build examples where hypotheses fail: dependent spanning lists, maps with same rank but not similar, non-diagonalizable matrices, and fields where a polynomial does not split.

Course-wide challenge bank

These mix proof and computation in the style expected from a theoretical linear algebra course.

Challenge 1 · Subspace

Let $W=\{p\in P_3(\mathbb{R}):p(1)=0\}$. Prove $W$ is a subspace and find a basis.

Solution sketch
Evaluation at $1$ is linear, so $W$ is its null space and hence a subspace. Since $p(1)=0$ iff $p(x)=(x-1)q(x)$ with $\deg q\le2$, a basis is $\{x-1,\ x(x-1),\ x^2(x-1)\}$.
Challenge 2 · Rank-nullity

If $T:P_2(\mathbb{R})\to P_2(\mathbb{R})$ is defined by $T(p)=p'$, find $\operatorname{rank}(T)$ and $\operatorname{nullity}(T)$.

Solution sketch
The null space consists of constant polynomials, so nullity is $1$. The image is all polynomials of degree at most $1$, so rank is $2$. Rank-nullity gives $2+1=\dim P_2=3$.
Challenge 3 · Determinant meaning

Prove that if $A$ and $B$ are similar matrices, then $\det A=\det B$.

Solution sketch
If $B=P^{-1}AP$, then $\det B=\det(P^{-1})\det(A)\det(P)=\det(A)$ because $\det(P^{-1})=1/\det(P)$.
Challenge 4 · Diagonalization

Is the linear map with matrix $A$ having rows $(2,1)$ and $(0,2)$ diagonalizable over $\mathbb{R}$?

Solution sketch
The only eigenvalue is $2$. The eigenspace solves $(A-2I)v=0$. Since $A-2I$ has rows $(0,1)$ and $(0,0)$, this means $v_2=0$ and the eigenspace is one-dimensional. A $2\times2$ matrix needs two independent eigenvectors to be diagonalizable, so $A$ is not diagonalizable.
04

Resources

MAT240H1 — official calendar listing
3Blue1Brown — Essence of Linear Algebra (intuition)

Companion text used by this guide: Friedberg, Insel & Spence, Linear Algebra. The public calendar confirms the MAT240H1 topic list but does not publish a current instructor textbook. Verify the assigned text, edition, and problem numbers against the active syllabus each year.