june-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 8 — June 3, 2026

Administrative

• Reflection 2: delayed due to personal circumstances. The delay will be compensated with a full reschedule — deadline will be set approximately one week from now (Wednesday of next week). No time is lost.
• TA and instructor office hours: polls were posted but very few students voted.
• Pre-midterm tutorial (week of June 9): practice problems will be posted Monday for everyone. During the tutorial, tell your TA which problem you want discussed as a group.
• Supplementary materials: going forward, additional written notes will be posted on Mondays to fill gaps left by the expositional style — proofs that were skipped, lemmas that are useful for the course, worked examples. These are course materials, not optional extras.
• PDF version of lecture notes is coming — useful for annotation on tablets.
• Midterm scope: linear transformations (this week’s material) are on the midterm. Material covered next week is not.

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1. What are we solving, and why does it matter?

Before writing down the definition of a linear transformation, it is worth asking: where does this concept come from? What is the question it answers?

The most basic kind of equation is for some function and some target value . Solving it means inverting : finding an such that maps to . For this to have a clean answer, the function needs to be injective — if two different inputs produce the same output, there is no unique solution and we cannot simply “undo” .

For example, the function is not injective on : both and satisfy . The “inverse” is only a partial inverse — it recovers one of the two solutions and quietly loses the other. By contrast, is injective on : different inputs always give different cubes, and recovers the unique solution correctly.

Now consider a system of linear equations. The input is a vector and the output is the vector of right-hand sides . The system defines a function . Solving the system means asking: for a given , which does send to ?

This function has a very special structure. Each output is just a weighted sum of the inputs, with fixed weights. This makes extraordinarily tractable — and the property that encodes this tractability is exactly linearity.

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2. Definition

Definition

Let and be vector spaces. A linear transformation (also called a linear map) is a function satisfying:

1. for all .
2. for all and .

These two conditions are often combined into one: for all and .

The word “transformation” is geometrically flavoured; “map” is more abstract. The words are synonyms, and both are synonyms for “function.” Different textbooks and subfields use them differently: calculus prefers “function,” algebra prefers “map,” geometry prefers “transformation.” In the context of this course, all three mean the same thing.

Why this definition? At first glance the two conditions seem arbitrary. The motivation from §1 makes them natural: for a system of linear equations, the function defined by the coefficients satisfies exactly these two properties — they follow from the distributivity of real multiplication. So every system of linear equations defines a linear transformation, and the abstract definition is designed to capture precisely this structure.

Automatic consequences

Two useful facts follow immediately from the definition. First, : setting in condition 2 gives . A transformation that does not send to cannot be linear. Second, : set .

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3. An infinite-dimensional example: the derivative

The definition is made abstract precisely so that it applies outside . The most important such example is differentiation.

The space . Define to be the set of all infinitely differentiable (smooth) functions : functions for which the derivative , the second derivative , and every higher derivative exist everywhere. With pointwise addition and scalar multiplication , this is a vector space.

Examples of elements: , , , all polynomials, and functions built by adding and multiplying these. Note that does not belong (its derivative is undefined at , let alone higher derivatives), and is in but not since its second derivative blows up at .

The differentiation operator. Define

This is well-defined: differentiating an infinitely differentiable function once still leaves it infinitely differentiable. Is it linear? The familiar rules from calculus say immediately:

These are exactly conditions 1 and 2. The differentiation operator is a linear transformation.

Why we can't write a matrix for

We showed in §5 below that every linear map is completely described by an matrix. This requires the input space to have a finite basis. The space is infinite-dimensional — no finite spanning set exists — so there is no matrix for . The abstract definition of a linear transformation is essential here: it gives us language for the derivative without requiring us to work inside .

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4. Linear maps are exactly matrices

For finite-dimensional spaces, the abstract definition and the coordinate description are equivalent.

Every matrix gives a linear map. Let be an matrix and define by . Then:

Both follow from distributivity of matrix multiplication. So every matrix-vector map is linear.

Every linear map gives a matrix. Conversely, suppose is linear. Let be the standard basis of . Any input decomposes as

Applying and using linearity:

Each is a fixed vector in — call its -th component . Then , and assembling these as columns gives the matrix with . The map is completely determined by where it sends the standard basis vectors.

Consequence. A linear map is determined by exactly numbers — the entries of its matrix. Knowing the matrix is exactly the same as knowing the map.

Definition

Given a linear map , the matrix of (with respect to the standard bases) is the matrix whose -th column is .

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5. The column picture of matrix-vector multiplication

The derivation above reveals an important alternative way to understand . Since and (the -th column of ):

The product is a linear combination of the columns of , with the entries of as coefficients.

This is different from the row picture introduced in Lecture 7, where the -th entry of was computed as the dot product of the -th row of with . Both descriptions produce the same vector — they are just two ways to organise the same arithmetic. The column picture is often more useful when thinking geometrically: it says that is always a specific linear combination of the columns, and as ranges over , the set of all outputs is the span of the columns of .

Column space

The set is called the column space of (also written or ). Its dimension is the rank of . The system has a solution if and only if lies in the column space of . Both definitions — via the image of the map and via the span of the columns — are the same thing.

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6. Examples

We work in and identify several familiar geometric operations as linear maps, recovering their matrices by the recipe of §4: apply the map to and , then stack the results as columns.

Rotation by counter-clockwise. The map rotates every vector counter-clockwise about the origin. Is it linear? Rotating two vectors and adding the results is the same as adding them first and then rotating — the parallelogram is preserved by rotation. So yes, it is linear.

To find the matrix: rotated gives . rotated gives . The matrix is

Applying it: .

Scaling. The map — stretch the -axis by and the -axis by — is linear (the formula is visibly linear in and ). , , so the matrix is . Diagonal matrices like this simply scale each coordinate independently.

Reflection about the -axis. . Matrix: .

Reflection about . — swaps the coordinates. Matrix: .

Identity. — does nothing. Matrix: .

Zero map. — sends every vector to . Matrix: .

Note on checking linearity

For maps defined by an explicit formula, checking linearity is easy: the formula must be linear in the inputs (no squaring, no cross-terms, no absolute values, no constants added outside). For maps defined geometrically, the check is usually more intuitive: rotations, reflections, and scalings all preserve vector addition and scalar multiplication visibly. When in doubt, both checks are available and can be used together.

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7. Composition and non-commutativity

Linear transformations can be composed: if and are linear, then the composition defined by is also linear. This corresponds to doing one transformation and then the other.

The order matters. In general, . This is already visible with ordinary functions: if and , then

These are not equal. For linear maps, composition corresponds to matrix multiplication — a product means “first apply , then apply .” We will define this precisely next lecture.

The non-commutativity of composition is geometrically transparent: rotate a vector and then reflect it about the -axis; then try the same operations in the opposite order. The two results are different.

Why matrix multiplication is "backwards"

The convention meaning “first , then ” comes from function notation: . Reading right to left matches the order of application, which is why the composition notation is written in the same backwards order. It is purely a notational convention, but it can cause confusion when first encountered.

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8. The matrix of a linear map between abstract spaces

So far we have worked with and and used the standard basis to extract a matrix. The same construction works for any finite-dimensional vector spaces, once bases are chosen.

Setup. Let be a linear map, where and . Choose a basis for and a basis for .

Decomposing the input. Any can be written uniquely as (by the unique representation proposition). Applying and using linearity:

Decomposing the output. Each is a vector in . Since is a basis for , we can write it uniquely as

Here (following the Einstein convention from Lecture 6) the upper index is the -component (row) and the lower index labels which basis vector is being sent (column).

The matrix. Assembling these coefficients, the matrix satisfies

where and are the coordinate vectors in the respective bases. The -th column of is the coordinate vector — exactly the same recipe as before, with the standard basis replaced by and .

The key point. A linear map between finite-dimensional spaces always has a matrix, once bases are chosen. The same map looks like different matrices in different bases. This is not a defect of the map — it reflects the fact that coordinates depend on the basis, while the underlying geometric transformation does not.

Why abstract spaces need this

A rotation inside a plane (a plane through the origin, not all of ) is a perfectly well-defined linear map . But is not ; it is a 2-dimensional subspace of a 3-dimensional space. To write a matrix for this rotation, you choose a basis of and proceed exactly as in §4 — the fact that sits inside is irrelevant to the internal structure of the rotation. The abstract definition handles this uniformly.

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Looking ahead

The next lecture will develop the algebra of linear maps and their matrices:

• Matrix multiplication: corresponds to composing the linear maps then .
• Null space and range: the preimage of and the image of the map, and the rank-nullity theorem relating their dimensions.
• Invertibility: when can a linear map be undone, and what does this require of the matrix?

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Textbook references

Topic	Axler (2nd ed.)	Hefferon (4th ed.)	Treil
Definition of linear map	Ch. 3, 3.1	Ch. 3, §IV.1	Ch. 3, §1
Linear maps = matrices	Ch. 3, 3.3	Ch. 3, §IV.2	Ch. 3, §1
Column picture of	Ch. 3, §3B	Ch. 3, §IV.2	Ch. 3, §1
Examples: rotations, reflections	Ch. 3, §3A	Ch. 3, §IV.1	Ch. 3, §1
Composition and non-commutativity	Ch. 3, §3D	Ch. 3, §IV.3	Ch. 3, §2
Matrix of a map between abstract spaces	Ch. 3, §3C	Ch. 3, §IV.2	Ch. 3, §2
and the derivative	Ch. 3 (exercises)	—	Ch. 3