MATH 332 — Fall 2026
Why Linear Algebra?
Diagnostic Survey due 8/21
October 9, 2026
We begin this semester amid significant change
Illinois Tech’s transformation
Painful departures of colleagues, mentors, and friends who have shaped this community; we grieve
New academic structures, no departments
Questions and uncertainty; let’s meet them with patience and care
We remain committed to your success
The loss is real, as is our responsibility to care for one another and keep learning well
The new structure connects depth, disciplines, and practice
- Depth in a chosen field remains essential
- Consequential problems cross traditional academic boundaries
- Interdisciplinary pathways should be easier to find and pursue
- Research, applied learning, and career preparation should connect more directly
Illinois Tech
↓
College of Tech Futures
↓
School of Computing
↓
(Applied Mathematics)
Technical depth + collaboration across fields + experience in practice
Further significant changes
The advance of AI
AI can be a capable assistant that helps solve research problems
- Ken Ono: How AI Will Unleash a New Kind of Intelligence
- Terence Tao: Mathematics in the Age of AI
- Leiden Declaration on Artificial Intelligence and Mathematics
- The Washington Post: Mathematicians ask what’s left for humans when AI can do math research
We need to understand what AI produces—and learn how to guide it
Politics
The political environment is also disruptive
How I use AI to support this course
- ChatGPT and Codex help me explore explanations, critique drafts, write and test code, and maintain course materials
- I choose the learning objectives, mathematical content, assessments, and grading standards
- I review and revise AI output and verify the mathematics, code, sources, and published materials
AI assists; responsibility remains mine
Course Map
Course decks
In this deck
🎬 Teaser Trailer
A Common Language for Representation and Action
image and data
transformations and constraints
networks and repeated change
data and approximation
systems of constraints
machine learning
Represent objects numerically
- Scalar \(a\) — one quantity
a = np.float64(22.5)- temperature
- price
- scale factor
- Vector \(\vct{x}\) — several related quantities
x = np.array([3, 2])- coordinates
- pixel values
- node scores
- system state
- Matrix \(\mat{A}\) — rectangular arrangement of quantities
A = np.array([[1, .5], [0, 1]])- data table or image
- relationships or constraints
- action on vectors
- Tensor \(\tens{T}\) — quantities indexed in several directions
T = np.zeros((32, 32, 3))- color image
- batch of matrices
- sequence of network states
Numerical representations are models of objects that allow us to compute with them
Act on these objects via transformations to obtain new objects
\(\vct{x}\)
\(\mat{X}\)
\(\mat{A}\)
\(\vct{y}=\mat{A}\vct{x}\)
\(\mat{Y}=\mat{A}\mat{X}\)
Examples of actions
- rotate or reflect
- stretch, scale, or compress
- shear
- project onto a lower-dimensional space
- change coordinate systems
- permute or select coordinates
- transform object coordinates to camera coordinates
- mix colors or features
- smooth, sharpen, or filter an image
- reduce dimension or extract features
- update a network state
- apply a Markov-chain transition
- average or combine measurements
Represent
An arrow is only the beginning
A vector can record
- a direction and displacement
- several measurements at once
- the current state of a system
- the ingredients of an object
\[ \vct{x}=\begin{bmatrix}3\\2\end{bmatrix} \]
A tiny image becomes numbers
a \(4\times4\) image
sampled pixel values
Choose an order; read off brightness
\[ \vct{x}=\begin{bmatrix} 17&51&221&238&34&119&\cdots&17 \end{bmatrix}^{\mathsf T} \]
\(4\times4\) image represented by a vector in \(\reals^{16}\)
Images contain important patterns
Each color channel is a matrix
\[\mat{M}=(m_{jk})_{j=1,k=1}^{m,n}\]
Once an image has been represented as a matrix, linear algebra can reveal important patterns
Singular value decomposition (SVD) decomposes a matrix into patterns ordered by their contribution
+ ···
Keep the most important patterns for compression, denoising, or discovering structure
Represent data for regression
observed pairs \((x_i,y_i)\)
Linear model
\[y_i\approx c_0+c_1x_i\]
One equation per observation
\[\begin{align*} c_0+c_1x_1&\approx y_1\\ c_0+c_1x_2&\approx y_2\\ &\ \vdots\\ c_0+c_1x_n&\approx y_n \end{align*}\]
The same system
\[ \mat{A}=\begin{bmatrix} 1&x_1\\ 1&x_2\\ \vdots&\vdots\\ 1&x_n \end{bmatrix}, \quad \vct{c}=\begin{bmatrix}c_0\\c_1\end{bmatrix}, \quad \vct{b}=\begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix} \]
\[\mat{A}\vct{c}\approx\vct{b}\]
Represent a network by an incidence matrix
Oriented edges
\[e_1:1\to2,\qquad e_2:2\to3,\qquad e_3:1\to3\]
Node–edge incidence matrix
\[ \mat{B}= \begin{bmatrix} -1& 0&-1\\ 1&-1& 0\\ 0& 1& 1 \end{bmatrix} \]
\(\mat{B}\) represents the network structure
- one row per node
- one column per edge
- \(-1\) at the tail; \(+1\) at the head
Act
A matrix is an action—predict before multiplying
\[ \mat{R}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}, \qquad \vct{x}=\begin{bmatrix}2\\1\end{bmatrix} \]
\[\vct{x}\longmapsto\mat{R}\vct{x}\]
\(\exstar\)
Before multiplying
- What geometric action does \(\mat{R}\) perform?
- Where does \(\vct{x}\) land?
- Then calculate \(\mat{R}\vct{x}\)
Composition is multiplication
\(\vct{x}_{\text{object}}\)
\(\vct{x}_{\text{world}}\)
\(\vct{x}_{\text{camera}}\)
\(\vct{x}_{\text{screen}}\)
place and orient the object in the world
express the world relative to the camera
project camera coordinates onto the screen
One composed action
\[ \vct{x}_{\text{screen}} =\mat{C}\mat{B}\mat{A}\vct{x}_{\text{object}} \]
Order of \(\left\{\begin{array}{c}\text{transformations}\\[-0.15em]\text{multiplication}\end{array}\right\}\) matters
What does the system tell us?
Three sensors; unknown location \((x,y)\)
\[\begin{align*} x+y&=6 &&\text{sensor 1}\\ 2x-y&=3 &&\text{sensor 2}\\ 3x&=9 &&\text{sensor 3} \end{align*}\]
All three constraints agree
Third equation is redundant: sum of the first two
More equations \(\ne\) more information
Does any solution satisfy every constraint?
Is there exactly one solution?
Which constraints add something new?
Can outputs determine the inputs?
Solving means understanding the information in a system—not only finding numbers
When the constraints cannot all be satisfied
No line passes through every point
So \(\mat{A}\vct{x}=\vct{b}\) has no exact solution
\[ \hvx =\argmin_{\vct{x}}\;\lVert\vct{b}-\mat{A}\vct{x}\rVert^2 \]
Least squares asks for the closest possible output
Projection identifies it geometrically
Repeated action reveals a persistent pattern
A page is important if important pages link to it
A different matrix—built from the network’s links—redistributes the node scores
\[ \vct{p}_{k+1}=\mat{A}\vct{p}_k \]
Repeated redistribution may approach a steady ranking
\[\mat{A}\vct{p}=\vct{p}\]
Eigenvectors identify directions that a matrix action preserves
Where machine learning fits
\(\vct{x}\)
learned affine layer
- \(\vct{x}\) — input features
- pixels, measurements, coordinates
- \(\mat{W}\) — learned transformation
- selects and mixes features
- \(\vct{b}\) — learned offset
- nonlinearity — activation
- ReLU (hockey stick), sigmoid
- prediction — model output
- class, score, estimated value
Linear algebra supplies representations, transformations, geometry, and scalable operations
Machine learning also needs probability, optimization, nonlinear models, data, and judgment
Compute Responsibly
From calculation aids to AI
Hidden Figures (2016)
1957 · FORTRAN 1972 · C 1978 · TeX 1978 · WORDSTAR 1983 · INTERNET (TCP/IP) 1984 · MATLAB 1991 · Python
Too little information, many answers
One measurement
\[ \begin{bmatrix}1&1\end{bmatrix}\vct{x}=4 \]
Many solutions
\[ \vct{x}=\begin{bmatrix}0\\4\end{bmatrix}, \begin{bmatrix}1\\3\end{bmatrix}, \begin{bmatrix}3\\1\end{bmatrix},\ldots \]
- existence: yes
- uniqueness: no
- information: insufficient
An ill-posed problem does not determine a unique, stable answer
Here uniqueness fails
One remedy: add a selection rule—e.g., the smallest-norm solution: \(\displaystyle\vct{x}_\star =\argmin_{[\,1\;1\,]\vct{x}=4}\lVert\vct{x}\rVert_2 =\begin{bmatrix}2\\2\end{bmatrix}\)
Almost parallel, almost unknowable
Each system has a unique solution
Nearly parallel constraints make that solution highly sensitive to the data
This is ill-conditioning
A tiny change in the data can cause a large change in the solution
Equivalent mathematics, unequal computations
Mathematically equivalent
\[ \sqrt{1+x^2}-1 = \frac{x^2}{\sqrt{1+x^2}+1} \]
For very small \(x>0\), direct subtraction combines nearly equal numbers
Catastrophic cancellation can discard significant digits
Python double precision:
| \(x\) | direct | rationalized |
|---|---|---|
| \(1.2346E-06\) | \(7.6206E-13\) | \(7.6208E-13\) |
| \(1.2346E-07\) | \(7.5495E-15\) | \(7.6208E-15\) |
| \(1.2346E-08\) | \(0.0000E+00\) | \(7.6208E-17\) |
The rationalized form avoids the subtraction
Numerical instability is avoidable error introduced or amplified by the computational method
A correct computation can still cost too much
Solving \(\mA \vx = \vb\) for \(\vx\)
For a dense \(n\times n\) matrix, \(\mA\)
- storage: about \(n^2\) numbers
- direct solve: about \(n^3\) operations
Double \(n\) implies
- \(4\times\) the storage
- \(8\times\) the arithmetic
Use reliable numerical linear algebra packages
- tested, numerically stable algorithms
- optimized libraries and available hardware
- specialized sparse and structured methods
Choose the efficient algorithm, e.g.
x=np.linalg.solve(A,b), notx=np.linalg.inv(A)@b
May reduce storage and operations by exploiting structure: sparsity, symmetry, low rank, …
Compute only what the problem requires
Unnecessary computation
- wastes time
- uses memory and energy
- creates additional opportunities for round-off error to accumulate or propagate
Examples:
- solve \(\mat{A}\vct{x}=\vct{b}\) directly instead of computing \(\mat{A}^{-1}\vct{b}\)
- reuse elimination structure or a factorization when many right-hand sides share \(\mat{A}\)
- exploit sparsity, symmetry, low rank, and other structure
Do no unnecessary computation
For production work, use a tested numerical library
Reliable software cannot fix ill-posedness or remove sensitivity inherent in an ill-conditioned problem
Can we trust the answer?
Does the mathematical problem determine an appropriate answer?
How much can the answer change when the data change slightly?
Does the computation introduce substantially more error?
Does the method use the size and structure of the problem efficiently?
Can residuals, another method, or known structure check the result?
Distinguish an ill-posed formulation, an ill-conditioned problem, an unstable algorithm, and excessive computational cost
The Course Ahead
Questions we will learn to answer
- How can we represent complicated objects and data?
- When does a system have one solution, many solutions, or none?
- Which information is genuinely new, and which is redundant?
- When can a transformation be reversed?
- How do we find the best approximate answer?
- Which patterns matter most?
Terms to know
Use this cumulative alphabetical index to revisit the first substantial treatment of each term
A–D
- Algebraic (Hamel) basis
- Angle between vectors, \(\theta\)
- Associativity, \((\mat{A}\mat{B})\mat{C}=\mat{A}(\mat{B}\mat{C})\)
- Augmented matrix, \([\,\mat{A}\mid\vct{b}\,]\)
- Back substitution
- Basis, \(\mathcal{B}\)
- Basis matrix, \(\mat{P}_{\mathcal{B}}\)
- Catastrophic cancellation
- Change of basis
- Column space, \(\operatorname{col}(\mat{A})\)
- Commutativity, \(\mat{A}\mat{B}=\mat{B}\mat{A}\)
- Composition of transformations
- Computational cost
- Consistent / inconsistent system
- Coordinate map, \(\vct{v}\mapsto[\vct{v}]_{\mathcal{B}}\)
- Coordinate vector, \([\vct{v}]_{\mathcal{B}}\)
- Covariance matrix, \(\mat{\Sigma}\)
- Cross product, \(\vct{u}\times\vct{v}\)
- Determinant, \(\det(\mat{A})\)
- Diagonal matrix, \(\mat{D}\)
- Dimension, \(\dim(V)\)
- Distance, \(\norm{\vct{u}-\vct{v}}\)
- Dot product, \(\vct{u}^{\mathsf T}\vct{v}\)
E–I
- Eigenvalue, \(\lambda\)
- Elementary matrix
- Elementary row operations
- Elimination
- Elimination matrix, \(\mat{M}_j\)
- Euclidean norm, \(\|\vct{v}\|_2\)
- Field, \(\reals,\ \mathbb{C},\ \mathbb{F}_2\)
- Finite-dimensional space
- Fourier mode, \(\phi_k(t)\)
- Frame
- Free variable
- Gauss–Jordan reduction
- Givens rotation, \(\mat{G}(i,j,\theta)\)
- Householder reflection, \(\mat{H}\)
- Hyperplane, \(\vct{a}^{\mathsf T}\vct{x}=b\)
- Identity matrix, \(\mat{I}\)
- Ill-conditioned problem
- Ill-posed problem
- Incidence matrix
- Inverse matrix, \(\mat{A}^{-1}\)
L–N
- Least squares
- Left null space, \(\operatorname{null}(\mat{A}^{\mathsf T})\)
- Leontief input–output model
- Linear combination, \(c_1\vct{v}_1+\cdots+c_k\vct{v}_k\)
- Linear independence
- Linear isomorphism
- Linear system, \(\mat{A}\vct{x}=\vct{b}\)
- Linear transformation, \(T\)
- Lower-triangular matrix, \(\mat{L}\)
- Matrix, \(\mat{A}\in\reals^{m\times n}\)
- Matrix–matrix multiplication, \(\mat{A}\mat{B}\)
- Matrix representation of a linear map
- Matrix–vector multiplication, \(\mat{A}\vct{x}\)
- Monomial
- Nonsingular / singular matrix
- Null space, \(\operatorname{null}(\mat{A})\)
- Nullity, \(\dim(\operatorname{null}(\mat{A}))\)
- Numerical instability
O–R
- Orthogonal complement, \(W^{\perp}\)
- Orthogonal decomposition
- Orthogonal matrix, \(\mat{Q}^{\mathsf T}\mat{Q}=\mat{I}\)
- Orthogonal vectors, \(\vct{u}^{\mathsf T}\vct{v}=0\)
- Parallel vectors, \(\vct{u}=c\vct{v}\)
- Permutation matrix, \(\mat{P}\)
- Pivot
- PLU factorization, \(\mat{A}=\mat{P}\mat{L}\mat{U}\)
- Polynomial space, \(\mathbb{P}_n\)
- Precision matrix, \(\mat{\Sigma}^{-1}\)
- Principal component analysis (PCA)
- Projection, \(\mat{Q}\)
- QR factorization, \(\mat{A}=\mat{Q}\mat{R}\)
- Rank, \(\operatorname{rank}(\mat{A})\)
- Rank factorization, \(\mat{A}=\mat{C}\mat{R}\)
- Rank–nullity theorem
- Reduced row-echelon form (RREF), \(\operatorname{rref}(\mat{A})\)
- Redundant constraint
- Reflection, \(\mat{H}\)
- Reversible transformation
- Rotation, \(\mat{R}_{\theta}\)
- Row space, \(\operatorname{row}(\mat{A})\)
- Row echelon form
S–Z
- Scalar, \(c\in\reals\)
- Scalar triple product, \(\vct{u}^{\mathsf T}(\vct{v}\times\vct{w})\)
- Singular value decomposition (SVD), \(\mat{A}=\mat{U}\mat{\Lambda}\mat{V}^{\mathsf T}\)
- Skew-symmetric matrix, \(\mat{A}^{\mathsf T}=-\mat{A}\)
- Solution / solution set
- Span, \(\operatorname{span}\{\vct{v}_1,\ldots,\vct{v}_k\}\)
- Standard basis, \(\{\vct{e}_1,\ldots,\vct{e}_n\}\)
- Standard matrix, \(\mat{A}\)
- Subspace, \(W\subseteq V\)
- Symmetric matrix, \(\mat{A}^{\mathsf T}=\mat{A}\)
- Tensor
- Total degree
- Transformation, \(T\) (matrix action: \(T_{\mat{A}}\))
- Transpose, \(\mat{A}^{\mathsf T}\)
- Tridiagonal matrix
- Upper-triangular matrix, \(\mat{U}\)
- Vector, \(\vct{v}\)
- Vector equation of a line, \(\vct{x}=\vct{p}+t\vct{v}\)
- Vector space, \(V\)
Complete the Diagnostic Survey
Complete Assignment 0 — Diagnostic Survey by August 21
Complete and submit this assignment individually, not as a group
How we will learn linear algebra
ask
represent
find an answer
understand
- calculate and visualize
- explain and prove
- experiment with Python in Jupyter/Colab notebooks
Computation supports understanding
It does not replace mathematical reasoning
Jupyter/Colab notebooks support demonstrations and experiments
Navigating this course
Use the slides for course content, the course website and repository for course materials, and Canvas for grades
These slides
Each slide deck has a list of decks and major sections after the title page
Each major section is headed by a slide that points to its minor sections
and jump to the previous and next major sections
and jump to the previous and next slides
« and » at the bottom jump to the previous and next decks
at the bottom left shows all slides, as does Esc
Course website, repository, and Canvas
-
Lecture slides are under the Schedule
Course work appears under Assignments and Quizzes and Tests
Basic course information is on the Welcome page
All materials are in the course repository, which builds the course website
Grades are in Canvas
Why participate synchronously?
- Keep pace with the course
- Ask questions and receive answers in real time
- Influence the pace and direction of discussion
- Help classmates learn and benefit from their questions
- Build a learning community and get to know one another
Using AI well
Roles overlap; choose deliberately. Think enough to know the problem AI should help solve
Tutor → build understanding
- Probe your understanding of the concept
- Reveal misconceptions with Socratic questions
- Contrast examples, nonexamples, and nearby concepts
- Connect intuition, definitions, formulas, representations
“Use Socratic questions, examples, and nonexamples to probe my understanding and connect intuition to the formal definition”
Critic / reviewer → test your work
- Find gaps, hidden assumptions, unclear reasoning
- Challenge a proof, derivation, program, or explanation
- Suggest tests or counterexamples without rewriting everything
“Find the weakest step in my argument and explain why”
Prefer dialogue: strengthen your thinking before producing finished work
Architect / collaborator → plan the work
- Decompose the problem into decisions and tasks
- Compare approaches, tradeoffs, failure modes
- Design an analysis, algorithm, program, workflow
“Propose two approaches, their assumptions, and how I could choose”
Contractor → execute a specification
- Implement code, plot, calculation, draft
- Best with clear inputs, requirements, checks, desired output
- Riskiest when you cannot evaluate the result
“Implement this specification and provide tests for each requirement”
Match tool to role: the best AI for one role may not be the best for another. One AI can help you reason and specify; another execute. You inspect and revise
Better-specified tasks permit more delegation
Responsibility never transfers to AI
Verify what AI produces
AI output: proposal to inspect, not authority. Expose assumptions, alternatives, reasons; verify appropriately
Mathematics
- Check definitions, hypotheses, theorem conditions
- Rework important steps; test examples and counterexamples
- Ask whether the conclusion follows from the stated assumptions
Computation
- Run the code and inspect the output
- Test edge cases, dimensions, units, signs, known special cases
- Confirm that the computation answers the intended mathematical question
Verify the result and whether the right problem was solved
Responsibility stays with you
Evidence and sources
- Inspect original sources for claims and citations
- Confirm that a source exists and supports the attributed claim
- Separate sourced facts from plausible inference
Communication
- Revise reasoning and wording yourself
- Explain each choice and answer questions about it
- Acknowledge meaningful assistance as required
Afterward: Can you explain the result, modify it, solve a related problem without reproducing AI output?
Delegate execution; retain understanding, judgment, responsibility
Next time: solving systems of equations
\[\begin{align*} x+y-z&=2\\ 2x-y+3z&=5\\ x+2y+z&=4 \end{align*}\]
- Does a solution exist?
- Is it unique?
- How can we solve it efficiently?
- What does the geometry tell us?
Everything in this course begins with questions like these
© 2026 Fred J. Hickernell · Illinois Tech · assisted by ChatGPT and Codex · Why Linear Algebra? · MATH 332 — Fall 2026 Website · \(\exstar\) = exercise