july-3

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 15 — July 3, 2026

Administrative

• Make-up midterm tomorrow: Saturday July 4, 9 AM, room IA2021 (a larger room, capacity ~60). All students who submitted a VOI or absence explanation should have received an email with confirmation. Format: 2 hours, 12 multiple-choice + 2 long-answer (instead of the original 3). Students who cannot attend in person: the PDF will be posted at noon; work on it independently for 2 hours.
• Reflection 2 late submissions: one course announcement had the wrong date (stated “Wednesday June 11th” when June 11th was actually a Thursday). Students who submitted Reflection 2 less than one day late will receive full credit for their submission without a late penalty.
• Reflection 3: not yet posted. All remaining reflections (6 total for the semester) will be scheduled at once so the timeline is predictable. Some may land later in the semester, which is not a bad thing — questions will be better aligned with the final. A full schedule will be posted.
• Regrade requests: always welcome. Submitting a regrade request, even on a marginal case, is never penalized. The process of preparing a regrade request forces you to review your work in depth, which is itself valuable.
• Final exam date: a preliminary schedule has been received from the department but is not yet ready to share — the date may change. It will likely fall in the second half of the exam session. Any potential conflicts with nearby CSC exams are being investigated.
• Office hours: the Google Calendar linked on Schedule and appointments now includes TA office hours — at least 4 hours per week, a mix of in-person and virtual. All hours are open to all students.
• Missed tutorials (June 30/July 1): makeup tutorial dates will be announced; both days have confirmed replacement slots later in the semester.
• Last lecture of the semester: the final Wednesday lecture will be held online (details to follow).

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1. Orientation and the three axioms — a second look

Before working with the determinant computationally, it is worth re-grounding the three axioms in the geometric intuition that motivated them.

Signed area in . The determinant is the signed area of the parallelogram spanned by and . If the pair is arranged counter-clockwise, the area is positive; if clockwise, it is negative. The key requirement is not which sign is assigned to which orientation, but that the two orientations get opposite signs. This is what makes the area function well-behaved (additive) rather than merely geometric (always non-negative).

Orientation in higher dimensions: the 4D god

A 2D creature (think: a figure on a sheet of paper) cannot convert a right-handed shape into its mirror image by moving within the plane — but a 3D being can: just lift the shape out of the plane, rotate it, and set it back down.

Exactly the same reasoning applies one dimension higher. A 3D being cannot turn a right shoe into a left shoe by any rotation or continuous motion within 3D space. But a 4D being could: lift the shoe into the fourth dimension, flip it, and return it. This is not merely science fiction — it is the precise content of the statement that orientation reversal in requires passing through .

What the determinant measures, in part, is whether a linear transformation is the “identity kind” (preserves orientation) or the “mirror kind” (reverses it).

The three axioms. For :

(N) Normalisation. .

(M) Multilinearity. Linear in each argument separately:

(A) Alternating (skew-symmetry). Swapping any two arguments changes the sign:

Theorem: there is exactly one function satisfying all three — the determinant.

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2. What happens if we drop normalisation?

If we keep (M) and (A) but drop (N), we no longer get a unique function. The zero function becomes a valid solution. But the freedom is much more limited than you might expect:

Theorem (Uniqueness up to scaling)

Suppose satisfies (M) and (A) only. Then for any matrix :

where is the identity matrix. In other words, is a scalar multiple of , and the scalar is simply .

The point: (N), (M), and (A) together force ; removing (N) leaves a one-parameter family of multiples .

Proof

The expansion in the standard basis (from Lecture 14) showed, for any satisfying (M) and (A):

The parenthesised sum is exactly , and .

On exam priority

The uniqueness proof — expanding into permutations — is conceptually the hardest part of this unit. It is important to understand why it works, but computation (cofactor expansion, recognising det = 0 ↔ not invertible) is more central to the final. If you are running short on study time, prioritise the applications.

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3. The case: a direct illustration

Let satisfy (M) and (A), and let , . Expand the first argument in the standard basis:

Expand the second argument in each term:

Using skew-symmetry :

This confirms: on a matrix equals . Under (N), , so , as required. Under (M) and (A) alone, we get the determinant scaled by — no other freedom exists.

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4. Computing a determinant — via column decomposition

Goal. Calculate without using any formula — reducing it entirely to determinants.

Step 1: decompose the first column by multilinearity. Write

By linearity in the first column:

Each matrix has a standard basis vector as its first column. The remaining columns, and , are fixed.

Step 2: apply column operations to simplify each term. By the proposition (proved in §5 below), adding a multiple of one column to another does not change the determinant. Use the standard basis vector in column 1 to zero out its row in the other columns:

• Matrix (I): first column . Add multiples of column 1 to columns 2 and 3 to zero out the entries:

• Matrix (II): first column . Zero out the entries in columns 2 and 3:

• Matrix (III): first column . Zero out the entries:

Step 3: identify each result as a determinant using the uniqueness theorem. Each simplified matrix has a standard basis vector in column 1; its determinant is a function of the block in the remaining rows and columns. That function is multilinear and alternating (inherited from the determinant). By §2, it equals for some scalar , which is found by plugging in for the block and computing the resulting determinant.

Term	Block	Identity substitution gives	det
(I)
(II)			(one swap)
(III)			(two swaps)

Step 4: assemble.

The structure behind the computation

The argument in Step 3 is not just a trick — it is the uniqueness theorem at work. We did not compute the determinants by brute force; we identified them as scalar multiples of determinants purely from their properties. The uniqueness theorem is doing real work here, not just providing a philosophical backdrop.

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5. Column operations and the determinant

The column operation used in Step 2 above deserves its own statement:

Proposition (Column operation type III)

For any vectors and scalar :

Proof

By multilinearity in the first column:

The second term vanishes because a matrix with two identical columns has determinant (swapping those columns changes the sign, but also returns the same matrix — so the determinant equals its own negative).

The full effect of column operations on :

• Swap two columns: changes sign.
• Scale a column by : scales by .
• Add (one column) to another: unchanged.

These correspond exactly to the three elementary column operations. The same results hold for row operations, because (the determinant of a matrix equals the determinant of its transpose) — swapping rows and columns plays the same role.

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6. Cofactor expansion — the general theorem

The computation in §4 generalises. Expanding along the first column:

Definition (Minor)

The (i,j)-minor of an matrix , written , is the determinant of the matrix obtained by deleting row and column from .

Theorem (Cofactor expansion along the first column)

The sign pattern for column 1 is starting at .

For a general column , the expansion is:

The sign pattern forms a checkerboard over the matrix:

Practical strategy. To expand along column : either use the formula directly with the signs, or swap column to the first position (introducing a sign flip) and then apply the column-1 formula.

Cofactor expansion along any row

The same formula works for rows:

This follows because . To expand along row 1, for example: For computations by hand, always choose the row or column with the most zeros to minimise work.

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7. Example: cofactor expansion applied directly

Expanding along the first column (entries ):

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8. The product formula

Theorem

For any matrices and :

The proof uses the uniqueness theorem: fix and consider . As a function of the columns of , is multilinear and alternating (since is linear in each column of ). By §2, . But , so .

Corollary. If is invertible, then and:

since forces .

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Textbook references

Topic	Axler	Hefferon	Treil
(M)+(A) implies	Ch. 4, 4.5	Ch. 4, §I.2	Ch. 4, §1
Column operations and det	Ch. 4	Ch. 4, §II.1	Ch. 4, §2
Minor and cofactor	Ch. 4, 4.7	Ch. 4, §II.2	Ch. 4, §2
Cofactor expansion	Ch. 4, 4.8	Ch. 4, §II.2	Ch. 4, §2
	Ch. 4, 4.9	Ch. 4, §II.4	Ch. 4, §2
	Ch. 4, 4.10	Ch. 4, §II.4	Ch. 4, §2