june-24

These lecture notes were generated by Claude inside a Cowork environment with knowledge of the syllabus and course materials, and generated from the audio transcript of the lecture. The lecture notes are not a transcript, and skip some parts while adding small details with sources/clarifications. These notes have been checked by me to be correct. I have added and edited the file for correctness and alignment with necessary material. All right-margin aside notes are additional to the course, and are not part of the learning objectives.

Lecture 13 — June 24, 2026

Administrative

• Midterm results: grades are being finalized (one long-answer question had a rubric miscommunication between instructor and TA, being corrected tonight). Results will be posted to Crowdmark as soon as this is resolved. The average is in the low 60s; the target was the low 70s — this is a calibration issue on the instructor’s part, not a reflection of the cohort’s understanding.
• Question 3C: nobody scored above 50%; nobody scored full marks. This question has been removed from the maximum. The new total denominator is 74 points (not a flat percentage bonus — 74 is the new max).
• Midterm statistics: approximately 10% of students scored ≥ 90. A normal distribution was observed, which is unusual for proof-based courses and suggests the questions tested understanding rather than proof-writing background.
• Make-up midterm: open to all students (details still being finalized with the department). Likely not graded for extra credit, but a practice opportunity in the same closed-book format. Covers material through this week.
• Regrade requests: submit as private Piazza questions within ~10 days of grade release. Details to follow in an announcement.
• Strong recommendation: spend at least 2 hours reviewing your midterm. More importantly, try to resolve the questions again on a blank piece of paper, without looking at anything. The gap between “I understand the solution when I see it” and “I can produce the solution” is where most learning happens.

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1. A linear map is fully determined by where it sends a basis

Proposition

Let be a linear transformation and let be a basis of . Then is completely determined by the values .

Proof

Let be any vector. Since is a basis of , there exist unique scalars such that

Applying and using linearity:

So is determined entirely by the values and the coordinates , which are in turn determined by .

This is why the matrix of — which records exactly where the basis vectors go — contains complete information about . A nonlinear function with tells us nothing about ; but a linear function with tells us , because is a basis of (as a vector space over itself).

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2. Matrix of a linear transformation in general bases

In Lecture 12 we worked exclusively in with standard bases. The same construction works for any vector spaces once bases are chosen.

Definition (Matrix of in chosen bases)

Let be a linear transformation. Choose a basis for and a basis for . The matrix of with respect to these bases is the matrix

whose -th column is the coordinate vector — the coordinates of expressed in the basis .

The dimensions are (dim ) × (dim ). Choosing different bases produces a different matrix for the same transformation.

Why this works in coordinates. Once bases are fixed, every vector in corresponds to a unique element of (its coordinates in ), and every vector in corresponds to a unique element of . The matrix is exactly the matrix that performs in these coordinate systems.

Choosing a basis is choosing an isomorphism

Fixing a basis of an -dimensional vector space builds a bijective linear map sending each to its coordinate vector . So “choosing a basis” and “identifying with ” are the same operation. After this identification, every abstract linear transformation becomes a matrix multiplication — which is why we can always reduce to column-vector arithmetic, regardless of what the original vectors “look like.”

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3. Example: the integration map

Let be defined by

i.e., the unique antiderivative of that equals at . The condition kills the constant of integration, making well-defined.

Linearity. and — both by standard integral laws.

Choosing bases. Take for and for .

Computing the matrix. Apply to each basis element, then read off coordinates in :

This matrix encodes the entire integration map in coordinate language. From now on, we can forget about polynomials and just multiply this matrix by coordinate vectors.

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4. Composition and the general matrix multiplication rule

If and have chosen bases (on ), (on ), (on ), then:

This is the same matrix multiplication formula derived in Lecture 12, now seen to work for arbitrary vector spaces rather than just .

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5. Inverse of a linear transformation

Definition (Inverse map)

Let be a linear transformation. A linear map is the inverse of if

has an inverse if and only if it is bijective.

The identity map and its matrix. The identity sends every vector to itself. In any basis of :

the identity matrix. This is because , whose coordinate vector in is .

Consequence on matrices. If for some basis , then is the matrix satisfying .

Lemma

If is a bijective linear transformation, then .

Proof

injective . By rank-nullity: . surjective . Therefore .

In particular, the matrix of any bijective linear map is square. The converse also holds: for equal-dimensional spaces, is bijective if and only if its matrix (in any basis) is invertible.

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6. Computing the inverse matrix

To find for an invertible matrix , we need the columns of . The -th column satisfies — a non-homogeneous linear system. All such systems share the same coefficient matrix , so we can solve them simultaneously:

The left block reduces to the identity matrix (this works precisely when is invertible — i.e., when there are no free variables, i.e., when ). The right block automatically becomes .

Example. Invert :

So .

When does this fail? If during row reduction a row of the left block becomes all zeros, it means has a free variable, i.e., , i.e., is not injective, i.e., is not invertible.

Solving linear systems. When is invertible, has the unique solution . This is the matrix analogue of “divide both sides by ” — which is now rigorously justified.

Connection to everything

The chain row-reduces to has no free variables is injective is bijective (since ) is one of the central equivalences of the course. All roads lead to the same condition.

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7. Determinants — first look

We have understood where a linear transformation sends things. Now we ask a geometric question: by how much does it scale areas and volumes?

Examples. Rotation does not change areas (). Scaling by in scales -dimensional volume by (a unit cube becomes a -cube). Projection onto a line kills all area ().

So we are looking for a function that takes a matrix and outputs the volume-scaling factor. What properties should it satisfy?

Thinking of the matrix as a family of column vectors (the images of the standard basis), the properties are:

1. Normalisation: (the identity doesn’t scale).
2. Multilinearity: is linear in each column separately:

3. Skew-symmetry: swapping any two columns negates the value:

Theorem (Existence and uniqueness of the determinant)

There is exactly one function satisfying properties (1), (2), and (3). It is called the determinant and denoted .

The proof (uniqueness from these properties alone) is the content of Lecture 14.

The formula. For :

Connection to invertibility: is not invertible iff its columns are proportional, i.e., , i.e., .

The formula (Sarrus’ rule mnemonic).

The three positive terms follow the main diagonal and two “wrapped” parallels; the three negative terms follow the anti-diagonal and its parallels.

Key connection.

The determinant is therefore a single-number test for invertibility, computable directly from the matrix entries.

Negative determinant

means the transformation reverses orientation: it includes a reflection somewhere. In , this means the transformation flips the plane over (swaps “clockwise” and “counterclockwise”). The magnitude still gives the volume-scaling factor.

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Textbook references

Topic	Axler	Hefferon	Treil
Matrix of in general bases	Ch. 3, 3.32–3.36	Ch. 3, §III.1	Ch. 2, §2
Integration as a linear map	Ch. 3 (examples)	Ch. 3, §III.2	Ch. 2
Inverse maps and bijections	Ch. 3, 3.51–3.56	Ch. 3, §IV.3	Ch. 3, §3
Matrix inverse via augmented matrix	—	Ch. 4, §IV.1	Ch. 3, §3
Determinant (introduction)	Ch. 4	Ch. 4, §I	Ch. 4, §1