may-13

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 3 — May 13, 2026

Administrative

• Piazza live Q&A is now active during lectures — use it to submit questions in real time instead of (or alongside) raising your hand.
• Tutorial format: the first tutorial ran this week. Submissions should reflect your genuine thinking about the problem — exploration, partial ideas, and questions all count. Rigorous proof-writing is not expected by default; clear reasoning is.
• Recordings: the lecture (only the lecturer) is audio-recorded, transcribed, and fed to Claude (with the full course folder as context) to produce these notes. The notes are not a transcript — they are a structured summary. They may contain errors; always cross-check with the textbook and your own notes.
• Review: plan to spend roughly as much time reviewing lecture material as the lecture itself lasted. A good review cycle is: compare your notes to these, read the textbook section, do a few exercises.
• Office hours: Wednesdays 1–3 pm, IC 3100 (Math & Stats Learning Centre).

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1. From one equation to two

The simplest equation is one variable, one condition:

There is exactly one solution: . Nothing interesting happens.

There are two natural directions to increase complexity. One is to keep one variable but make the equation harder — quadratic, polynomial, transcendental. That is the world of calculus. The other is to keep the equation linear but add more variables:

What does it mean to solve this? Not to find a solution, but to describe all solutions. The equation has two variables, so a solution is a pair . The pair works: . But so does , and , and infinitely many others.

To describe all solutions, isolate one variable in terms of the other and introduce a parameter:

For every real number , this gives a solution . And every solution arises this way. This is the parametric form of the solution set.

Why not just list solutions?

Because the solution set is infinite — you cannot enumerate it. A parametric description is a recipe: it tells you how to generate all solutions from a single free parameter. As systems get larger (many variables, many equations), parametric descriptions become even more essential.

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2. The solution set is a line

Since we have two variables, we can draw the solution set on the -plane. The equation defines a straight line.

Why a line, and not something more complicated? Here is one way to see it.

Take any two solutions and . Both satisfy the equation:

Consider the vector from to :

Subtract the first equation from the second:

The displacement vector satisfies the same equation, but with zero on the right. This tells us:

The ratio of the two components of is always , regardless of which two solutions we pick. So all displacement vectors along the solution set are proportional to a single direction — which is the definition of a straight line. The direction vector is, for example,

since .

The slope as a vector ratio

In high-school terms, is the slope of the line . In the vector language we are building, slope is just the ratio of the second component to the first component of the direction vector. The vector viewpoint generalises to higher dimensions where “slope” has no direct meaning but direction vectors do.

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3. Homogeneous equations

Definition. A linear equation is homogeneous if its right-hand side is zero:

An equation with a nonzero right-hand side is non-homogeneous.

In the example above, the original equation is non-homogeneous. The equation satisfied by the direction vectors is — homogeneous.

Geometric picture. The solution set of is a line that passes through (and many other points), but not through the origin. The solution set of is a parallel line that does pass through the origin. The direction vectors of the first line are exactly the solution vectors of the second.

Can we add solutions of ? If and are both solutions, their sum satisfies

The sum is not a solution of the original equation — it gets kicked off the line. The “5” accumulates.

But for the homogeneous equation? If and both satisfy , then

The sum stays on the line. This is the key observation, and it generalises to any homogeneous equation in any number of variables.

Particular solution + homogeneous

The full solution set of the non-homogeneous system can always be written as

where is any single solution of the original system, and ranges over all solutions of the associated homogeneous system. This decomposition is analogous to the general solution of a linear ODE (a particular solution plus the homogeneous solution). It is proved rigorously in Hefferon, Ch. 1, §I.3, Lemma 3.7.

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4. Lemma: sums of homogeneous solutions

Lemma. Given a homogeneous linear equation, the sum of any two solutions is again a solution.

Proof. Let the equation be

where are fixed. Suppose and are both solutions. Then:

Adding (1) and (2) and grouping by coefficient:

This says that satisfies the equation.

Why does homogeneity matter? If the right-hand side were , adding two equations would give . The sum would solve a different equation, not the original. Homogeneity — having zero on the right — is what makes close the argument.

The same lemma applies equation-by-equation to any system of homogeneous equations: if and each satisfy every equation in the system, so does .

This was the first proof of the course.

It was a complete, rigorous argument with no gaps. Notice its structure: (1) state what it means for the vectors to be solutions, (2) perform an algebraic manipulation, (3) recognise the result as the definition of the sum being a solution. This template — unpack, manipulate, repack — recurs throughout linear algebra.

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5. Three ways to read a system of equations

Consider the system in three variables:

This is a single mathematical object, but it admits three genuinely different geometric interpretations. All three are correct; each makes different aspects of the structure visible.

Interpretation 1 — Linear combinations

Rewrite the system as:

The question becomes: for what scalars is the right-hand vector expressible as a linear combination of the three column vectors? The scalars are the “amounts” of each column vector you take.

This is the column picture. It will become central when we study span and linear independence.

Interpretation 2 — Dot products and perpendicularity

Each equation can be read as a dot product:

Recall that the dot product of two vectors and satisfies

where is the angle between them. In particular, if and only if .

For the homogeneous version of each equation (right-hand side zero), must be perpendicular to the corresponding row vector. In , the set of all vectors perpendicular to a single nonzero vector is a plane through the origin. The solution set of the homogeneous system is the intersection of two such planes — generically, a line through the origin.

This is the row picture. It will reappear when we study orthogonality and inner products.

What is the dot product, really?

The formula defines the dot product computationally. The geometric formula defines it in terms of angle. That the two agree is a nontrivial theorem (a consequence of the law of cosines). The dot product will be generalised significantly in Chapter 6 of Axler, where it becomes the inner product — an abstraction that allows us to measure angles and lengths in any vector space, not just .

Interpretation 3 — Linear transformation

Rewrite the system compactly as a matrix-vector equation:

The matrix encodes a function : it takes a three-dimensional input and produces a two-dimensional output by the rule prescribed by the six entries. Solving the system is asking: which input vectors map to the specific output ?

This is the transformation picture, and it is the most powerful of the three. It will be developed fully when we study linear maps, null spaces, and the rank-nullity theorem.

Gaussian elimination

The mechanical technique for solving systems of linear equations — eliminating one variable at a time by adding multiples of one equation to another — is called Gaussian elimination. It will be covered in detail in the computational portion of the course. For now, the focus is on interpreting what a solution set is, not on how to find it algorithmically. See Hefferon, Ch. 1, §I.1–I.2 for a careful treatment.

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6. New examples of vector-like structures

The lemma from §4 shows that solution sets of homogeneous equations behave like vectors: they can be added, and scalar multiples of solutions are again solutions. But this behaviour is not unique to . Here are two more examples where addition and scalar multiplication arise naturally.

Sequences

A real sequence is a function , assigning a real number to each natural number. We write the values as , with the subscript recording the position.

Example — Fibonacci. Define by , , and for :

Operations on sequences. Given two sequences and , define:

• Sum: — add term by term.
• Scalar multiple: — scale every term by the same number.

These are exactly the same operations as on vectors in , but the “components” are now indexed by rather than . A sequence is, in a precise sense, an infinite-dimensional version of a vector.

in Axler

Axler (Ch. 1, “Definition of Vector Space”) introduces — the set of all infinite sequences of elements of — immediately after , with addition and scalar multiplication defined term by term. He verifies that is a vector space. This is exactly the structure described here for real sequences.

Polynomials

A polynomial with real coefficients is a function of the form

The number is the degree of , written ; is the leading coefficient.

Operations. Addition of polynomials is defined pointwise: . Scalar multiplication: .

Example. Taking polynomials of degree (quadratics):

A subtlety. If we restrict to polynomials of exactly degree 2, the set is not closed under addition: and are both degree 2, but their sum is , which has degree 0. To get a set that is closed, we take all polynomials of degree at most . Adding two such polynomials can only lower the degree, never raise it.

Scaling a polynomial of degree by a nonzero scalar keeps it at degree . Scaling by zero gives the zero polynomial — which we allow (it has no degree, or by convention degree ).

Functions in general

More generally, if is any set of real-valued functions on a common domain, we can define and . The sum and scalar multiple are again functions in (as long as is closed under these operations).

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7. Abstract vector spaces

The examples above — , solution sets of homogeneous systems, sequences, polynomials, functions — all share two operations that behave in the same way. This calls for a general definition.

Fields and scalars

In this course, scalars are real or complex numbers. But the definition of a vector space works equally well if scalars come from any field — a set with addition and multiplication satisfying the usual rules (commutativity, associativity, distributivity, inverses). The complex numbers form a field, as do the rationals . Finite fields (e.g., for prime ) also give useful vector spaces. We will switch to as our field of scalars when we discuss eigenvalues, because polynomials over always have roots. Axler handles both cases simultaneously by writing throughout, where is either or . Definition. A real vector space is a set equipped with two operations:

satisfying the following axioms, for all and all :

	Axiom
(A1)	(commutativity of addition)
(A2)	(associativity of addition)
(A3)	There exists such that for all (zero vector / additive identity)
(A4)	For each there exists with (additive inverse)
(S1)	(multiplicative identity)
(S2)	(associativity of scalar multiplication)
(D1)	(distributivity over vector addition)
(D2)	(distributivity over scalar addition)

Elements of are called vectors; elements of acting on are called scalars.

Why this list? Each axiom is needed. Remove any one, and familiar conclusions fail: without (A3) you cannot define subtraction; without (A2) you cannot reorder long sums; without (D1) you lose the distributive law that makes algebra work. A useful exercise is to try to prove something simple (e.g., ) and observe which axioms get used.

Groups

If you drop scalar multiplication entirely and keep only the addition axioms (A1)–(A4), you get a structure called an abelian group. Groups are one of the most fundamental notions in all of mathematics — they arise in number theory, geometry, cryptography, and physics (symmetry groups of physical laws). Vector spaces are abelian groups with extra structure: a compatible scalar multiplication. A first course in abstract algebra would study groups; this course studies groups-with-scalars.

The examples revisited. Each of the following is a vector space with the operations described:

• (componentwise and scalar multiplication)
• The solution set of any homogeneous linear system in (a subspace of — a subset that is itself a vector space under the inherited operations)
• : sequences, with termwise operations
• : polynomials of degree at most
• : all polynomials
• Any set of functions closed under pointwise addition and scalar multiplication

Verifying that a given set is a vector space means checking all eight axioms. In practice, many of them are inherited automatically from (commutativity, associativity, distributivity), so the most important conditions to check are closure: that and stay inside the set, and the existence of a zero vector.

Subspaces

A subset that is a vector space under the same operations is called a subspace of . To verify a subspace, you do not need to re-check all eight axioms — just three: (i) , (ii) is closed under addition, (iii) is closed under scalar multiplication. Axioms (A1), (A2), (D1), (D2), (S1), (S2) are automatically inherited. The solution sets of homogeneous systems are the archetypal examples of subspaces.

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Looking ahead

The next step is to develop a working vocabulary for vector spaces: which vectors can be reached by taking linear combinations, what it means for a set of vectors to be independent, and what a basis is. These are the concepts in Chapter 2 of Axler and Chapter 2 of Hefferon.

Gaussian elimination — the algorithm for solving systems of linear equations — will also be developed, completing the story begun today in §5.

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Textbook references

Hefferon's order of development

Hefferon is the textbook best aligned with this lecture. It introduces linear systems and homogeneous equations before defining vector spaces, so that the abstract definition in Chapter 2 arrives motivated by everything in Chapter 1. Reading Chapter 1, §I.1–I.3 of Hefferon directly after today’s lecture will feel like a careful written version of what was discussed here. Axler, by contrast, starts immediately with abstract vector spaces and introduces sequences and polynomials as examples in Ch. 1 — no linear systems at all. Both approaches are valuable; together they give a fuller picture.

Nota bene

Beware the reference section. Although several measures have been done to prevent hallucinations, typos are sometimes possible in this section. I (Denis, just in case) usually double-check the references, but I might miss some mistakes.

Topic	Axler (2nd ed.)	Hefferon (4th ed.)
Linear equations (one and two variables)	—	Ch. 1, §I.1
Parametric solution sets	—	Ch. 1, §I.2
Homogeneous equations — definition	—	Ch. 1, §I.3, Def. 3.2
Sums of homogeneous solutions (our lemma)	—	Ch. 1, §I.3, Lemma 3.6
General solution = particular + homogeneous	—	Ch. 1, §I.3, Lemma 3.7
Linear combinations	Ch. 2, “Span and Linear Independence”	Ch. 1, §I.1
Dot product (preview)	Ch. 6, “Inner Products”	Ch. 3, §VI
Matrix-vector notation, linear maps (preview)	Ch. 3, “Definitions and Examples”	Ch. 3, §I.1
Abstract vector space — definition and axioms	Ch. 1, “Definition of Vector Space”	Ch. 2, §I.1, Def. 1.1
Sequences as a vector space ()	Ch. 1, “Definition of Vector Space”	Ch. 2, §I.1, Example 1.14
Polynomials as a vector space ()	Ch. 1, “Definition of Vector Space”	Ch. 2, §I.1, Example 1.5
Subspaces — definition and tests	Ch. 1, “Subspaces”	Ch. 2, §I.2, Def. 2.1