may-29

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 7 — May 29, 2026

Administrative

• Reflection 2: will be posted today or Monday. If delayed to Monday, the deadline shifts from Wednesday to Friday — you will not lose time.
• Tutorial 2 feedback: graded submissions are now in your OneDrive folders. A few students received the wrong variant; if yours looks incorrect, contact the instructor.
• Midterm: Saturday, June 13, 9–11 am.
• Pre-midterm tutorial: in the week of June 7, tutorial problems will be posted in advance on Monday (mock midterm style). During the tutorial itself, choose a problem you’d like discussed as a group. This format replaces the usual scaffolded problem set for that week.
• TA office hours: TAs will be assigned in-person office hours starting next week, clustered around tutorial times. An online poll for additional instructor office hours will also be posted.

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1. Unique representations in a basis

Last lecture ended with the definition of dimension and the theorem that all bases of a finitely-spanned vector space have the same size. A corollary of that machinery is the following central proposition, which is the real payoff of having a basis.

Proposition

Let be a vector space of dimension and let be a basis of . For every there exist unique scalars such that

Proof

The two parts of the statement — existence and uniqueness — come from the two parts of the definition of a basis.

Existence. Since is a basis, it spans . By the definition of span, every vector in is a linear combination of .

Uniqueness. Suppose has two representations:

Subtracting the right-hand side from the left:

Since are linearly independent, the only vanishing linear combination is the trivial one. Therefore for all , meaning the two representations coincide.

This proposition is why bases are useful: once you have fixed a basis, every vector is encoded by exactly one list of numbers. Change the basis, and the same vector is encoded by a different list of numbers — but the encoding is still bijective and unambiguous.

The connection back to Lecture 6

The uniqueness argument above is exactly the contrapositive of the lemma from §2 of Lecture 6: a set of vectors is linearly dependent if and only if some vector can be written as two different linear combinations of them. Rephrased: a set is linearly independent if and only if every vector in its span has a unique representation. This is why independence is the right condition for a basis.

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2. Coordinates

The scalars in the proposition are called the coordinates of with respect to the basis . We assemble them into a column vector called the coordinate vector.

Definition

Let be a basis of and let . The coordinate vector of with respect to is

where are the unique scalars satisfying .

The proposition guarantees that is well-defined: the coordinates exist and are unique.

Example. We work in with three different bases and the vector .

• Standard basis . We have , so .

• Diagonal basis . We need , giving the system , , so , . Thus .

• Sheared basis . We need , giving , , so . Thus .

The same geometric object is described by three different lists of numbers — and in the case, the coordinates happen to coincide with the components of in , which is only because is the standard basis.

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3. Dimension as a shortcut for recognising bases

Checking that a set is a basis requires verifying two conditions: it spans , and it is linearly independent. But dimension gives us a powerful shortcut.

Claim. In a vector space of dimension , any set of exactly linearly independent vectors is automatically a basis (and dually, any set of exactly vectors that spans is automatically linearly independent).

Why? Suppose are linearly independent in and suppose for contradiction they do not span . Then there exists a vector . Adding to the set gives linearly independent vectors in . But then by the Steinitz lemma, any spanning set of must have at least elements — contradicting . So must already span .

Applied to the example. Is a basis of ? We verify linear independence by solving : this gives and , so . Since we have linearly independent vectors in a 2-dimensional space, spanning follows automatically — no separate check needed.

The dual shortcut

The argument runs symmetrically from spanning: if vectors span but are linearly dependent, one can be removed (by §4 of Lecture 6) to give a spanning set of size . Any linearly independent set would then have at most elements — but has a basis of elements, contradiction. So vectors spanning must be independent.

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4. Bases as coordinate grids

A useful geometric picture: once you fix a basis , you are choosing a grid on . The coordinate vector records how many steps you take along each grid direction to reach .

For the three bases in the example above:

• gives the standard axis-aligned grid: steps east and north.
• gives a grid along the diagonal directions and : steps along and .
• gives a sheared grid: one axis is horizontal, the other is at .

The vector is the same point in all three grids. What changes is how you walk there. There is nothing intrinsically better about : it is preferred only because it is the most numerically transparent, not because it carries more geometric information.

This perspective — a basis as a choice of coordinate system — is what the phrase change of basis refers to. Given two bases and , how do you convert to ? This will be addressed in a future lecture.

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5. Infinite-dimensional spaces

Everything above assumed is finite-dimensional. As a cautionary example, we examine what happens when it is not.

The space of all real sequences. Define

Equivalently, is the set of all functions , where is the -th term (this is an exercise!). With component-wise addition and scalar multiplication, this is a vector space.

An important subspace. Let

be the subspace of eventually-zero sequences: each sequence may have finitely many nonzero terms, but the cutoff point depends on the sequence. For example, , , and all belong to , while does not.

Checking: is closed under addition (the max of two cutoff points is a cutoff for the sum) and scalar multiplication, and contains . So it is indeed a subspace.

A basis for . Define the standard sequence vectors

with having a in position and s elsewhere. The (infinite) set is a basis for :

• Spanning. Any equals , which is a finite linear combination of the .
• Independence. If for any finite collection, looking at the -th component gives .

This is the first example of an infinite basis: a basis that spans and is independent, but has infinitely many elements. The proposition from §1 still holds: every element of has a unique representation as a linear combination of the — but now the representation involves a finite sum (which is what “linear combination” always means) from an infinite list.

What goes wrong in . The sequence belongs to but not to . No finite linear combination of the produces , so does not span .

In fact, has no explicitly constructible basis. Bases of (called Hamel bases) do exist — but their existence requires the Axiom of Choice, an additional set-theoretic assumption beyond the usual axioms. No explicit list of vectors that spans and is linearly independent can be written down.

The Axiom of Choice and Banach-Tarski

The Axiom of Choice says, roughly, that given any collection of non-empty sets you can pick one element from each, even with infinitely many sets and no rule for how to choose. This seems innocuous, but in very large infinite settings it leads to counterintuitive consequences. The most famous: using the Axiom of Choice, one can prove the Banach-Tarski paradox — a solid ball in can be decomposed into finitely many pieces and reassembled, using only rotations and translations, into two solid balls of the same radius. The pieces involved are not measurable (they have no well-defined volume), which is why physical intuition breaks down. The existence of Hamel bases for is another such consequence.

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6. Matrix-vector multiplication

The last few minutes of lecture introduced the notation that will be central to the rest of the course.

Recall how we encoded a system of linear equations using an augmented matrix. Given the system

we associated the coefficient matrix and the column vectors , . The system can then be written in shorthand as .

This shorthand is only meaningful if we define what means.

Definition

Let be an matrix with entries (row , column ) and let . The matrix-vector product is the vector in whose -th entry is

In other words, the -th entry of is the dot product of the -th row of with the vector .

Why write it this way? The definition is motivated by the system above: the left-hand side of the -th equation is exactly . Writing is thus a compact encoding of the entire system.

The notation also inherits a key algebraic property: matrix-vector multiplication is linear:

These follow directly from the definition and the distributivity of real multiplication. The system is the homogeneous system, and its solution set is the subspace we have been studying.

Matrix-vector multiplication and linear maps

The map is a function from to that is linear: it preserves addition and scalar multiplication. Conversely, every linear map between finite-dimensional spaces can be described by a matrix in this way, once bases are chosen. This connection — between matrices and linear maps, between coordinates and coordinate-free geometry — is the central theme of the second half of the course.

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Looking ahead

With coordinates and matrix-vector multiplication in hand, the next lectures will:

• Define linear maps abstractly and show they are in bijection with matrices (once bases are fixed).
• Develop the algebra of matrices: matrix-matrix multiplication, transpose, inverse.
• Study the rank of a matrix as the dimension of the column space, and prove the rank-nullity theorem.

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Textbook references

Topic	Axler (2nd ed.)	Hefferon (4th ed.)	Treil
Unique representation in a basis	Ch. 2, 2.29	Ch. 2, §I.1, Thm. 1.12	Ch. 2, §3
Coordinates and coordinate vectors	Ch. 3, 3.1	Ch. 2, §III	Ch. 2, §3
independent vectors in -dim space span	Ch. 2, 2.39	Ch. 2, §I.1	Ch. 2, §3
Infinite-dimensional spaces, Hamel bases	Ch. 2 (remarks)	—	—
Matrix-vector multiplication	Ch. 3, §3B	Ch. 3, §IV.1	Ch. 3, §1