U of T · Algebra II · Winter

MAT247H1: length, angle, and canonical form.

MAT240H1 asked what a linear map does. This course adds geometry — an inner product gives vectors length and angle — and then finishes the classification MAT240H1 started: every operator has a canonical form, even the ones that refuse to diagonalize.

About this course

Where linear algebra becomes geometry.

MAT247H1 "Algebra II": real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations, the adjoint, Hermitian and symmetric transformations, the spectral theorem for symmetric and normal transformations, the polar representation theorem, the primary decomposition theorem, rational and Jordan canonical forms, and additional topics including dual spaces, quotient spaces, bilinear forms and quadratic surfaces. Follows MAT240H1; co-requisite MAT158H1 + MAT159H1. This guide uses Friedberg, Insel & Spence as a companion text.

01

Chapters

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

MAT247H1Algebra II · Winter MAT240H1Algebra I · prerequisite Co-reqMAT158H1 + MAT159H1

  1. Inner product spaces

    Real and complex inner products, norms, Cauchy–Schwarz.

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  2. Orthogonality & Gram–Schmidt

    Orthonormal bases, orthogonal complements, projections.

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  3. The adjoint

    Riesz representation, $T^*$, and least squares.

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  4. Isometries: orthogonal & unitary operators

    Maps that preserve length, and what they look like.

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  5. Hermitian, symmetric & normal operators

    Self-adjointness and the normal condition $TT^*=T^*T$.

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  6. The spectral theorem

    Orthonormal eigenbases for normal and symmetric operators.

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  7. Polar & singular value decomposition

    $T=UP$, and the singular values of any map.

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  8. Dual spaces & quotient spaces

    Functionals, annihilators, and $V/W$.

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  9. Primary decomposition & Jordan form

    Generalized eigenvectors and the Jordan canonical form.

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  10. Rational form, bilinear & quadratic forms

    Canonical form over any field; quadratic surfaces.

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Textbook Spine

The public calendar lists MAT247H1's topics in the order used here. This guide uses Friedberg, Insel & Spence, Linear Algebra, as the companion text, because its chapters on inner product spaces and canonical forms follow the same route. Check the current syllabus for the exact edition and assigned problems.

Unit-to-textbook alignment
Geometry arrives

Units 1-3 pair with the text's inner product space chapter: inner products, norms, orthonormal bases, Gram–Schmidt, orthogonal complements, and the adjoint. This is the material with no MAT240H1 analogue at all.

Operators with structure

Units 4-7 pair with the sections on isometries, normal and self-adjoint operators, the spectral theorem, and the polar and singular value decompositions. The theme: an extra hypothesis buys an orthonormal eigenbasis.

Classification without geometry

Units 8-10 pair with dual spaces, quotients, and the canonical form chapters. Here the inner product goes away again and the question is what an operator looks like over any field.

CONJUGATE

Watch which slot

Complex inner products are linear in one argument and conjugate-linear in the other. Fix your convention on day one and never drift; half of all MAT247 sign errors start here.

ADJOIN

Translate to $T^*$

Most theorems in this course are a statement about how $T$ and $T^*$ interact. Before proving anything, write the hypothesis as an equation relating them.

DECOMPOSE

Split the space

Spectral theorem, primary decomposition, and Jordan form are all the same instinct: break $V$ into pieces the operator preserves, then look at it one piece at a time.

COMPARE

Real versus complex

Over $\mathbb{C}$ every normal operator diagonalizes; over $\mathbb{R}$ you need symmetry. Keep a running list of which theorems need which field.

03

Practice Plan

MAT247 rewards knowing which structural hypothesis you are standing on. The unit pages give guided questions; use this course-level mix to build exam readiness.

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Hypothesis tracking

For every theorem, write down whether it needs an inner product, needs $\mathbb{C}$, or needs self-adjointness. Most exam errors are a theorem applied one hypothesis short.

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Gram–Schmidt by hand

Run the algorithm on small explicit sets until it is automatic, including in a function space with an integral inner product.

03

Canonical form drills

Given a characteristic and a minimal polynomial, list every possible Jordan form. This is the single most testable computation in the course.

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Counterexample sets

Keep examples of: a normal operator that is not self-adjoint, a real matrix with no real eigenvalue, and two matrices with the same characteristic polynomial but different Jordan forms.

Course-wide challenge bank

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

Challenge 1 · Cauchy–Schwarz sharpness

Prove $|\langle u,v\rangle|\le\|u\|\,\|v\|$ with equality exactly when $u$ and $v$ are dependent.

Solution sketch
If $v=0$ both sides vanish. Otherwise put $c=\langle u,v\rangle/\|v\|^2$ and expand $0\le\|u-cv\|^2=\|u\|^2-|\langle u,v\rangle|^2/\|v\|^2$. Equality forces $u-cv=0$, so $u$ is a multiple of $v$.
Challenge 2 · Adjoint of a projection

Let $P$ be the orthogonal projection onto a subspace $W$. Prove $P^*=P$ and $P^2=P$.

Solution sketch
Write $u=u_W+u_{W^\perp}$. Then $\langle Pu,v\rangle=\langle u_W,v_W\rangle =\langle u,Pv\rangle$, so $P^*=P$; and $Pu\in W$ already, so $P(Pu)=Pu$.
Challenge 3 · Normal but not self-adjoint

Give a normal operator on $\mathbb{C}^2$ that is not self-adjoint, and confirm the spectral theorem still applies.

Solution sketch
Take $A=\left(\begin{smallmatrix}i&0\\0&-i\end{smallmatrix}\right)$. Then $A^*=-A$, so $AA^*=A^*A=I$ and $A$ is normal but $A^*\ne A$. It is already diagonal with eigenvalues $\pm i$ — non-real, which is exactly why it is not self-adjoint.
Challenge 4 · Jordan from polynomials

An operator on a $4$-dimensional space has characteristic polynomial $(t-3)^4$ and minimal polynomial $(t-3)^2$. List the possible Jordan forms.

Solution sketch
Blocks all have eigenvalue $3$, the largest is $2\times2$ (minimal polynomial), and sizes sum to $4$. So either $2+2$, or $2+1+1$. Two possibilities; the eigenspace dimension ($2$ or $3$) tells them apart.
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Resources

MAT247H1 — 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 MAT247H1 topic list but does not publish a current instructor textbook. Verify the assigned text, edition, and problem numbers against the active syllabus each year.