Zullo Linear Algebra Tutorbot is a study assistant designed for students attending Prof. Ferdinando Zullo’s Linear Algebra course.
It can help you:
review definitions, theorems, and course topics;
understand mathematical concepts;
practise exercises;
check your solutions;
identify common mistakes;
prepare for written and oral examinations.
Zullo Linear Algebra Tutorbot follows the terminology, notation, methods, and topics used in the course lecture notes and exercise sheets.
Important: Zullo Linear Algebra Tutorbot is a study-support tool. Official course announcements and instructions from Prof. Zullo or the University always take precedence.
Command
Function
/hint
Receive one hint without the complete solution
/solution
Receive a complete, step-by-step solution
/oral
Start an oral-examination simulation
/written
Generate a written mock examination
/quiz
Start an interactive quiz
/difficulty easy
Select basic questions and direct calculations
/difficulty medium
Select standard exercise-sheet questions
/difficulty hard
Select examination-level problems
You can combine commands with topics or specific instructions.
For example:
/oral Ask me about bases and dimension.
/difficulty hard
/written Create a mock examination on linear systems and diagonalization.
You can ask questions using ordinary language.
What is the difference between linear independence and orthogonality?
How can I determine whether three vectors form a basis of (\mathbb{R}^3)?
Can you explain the Rouché–Capelli theorem?
Why is matrix multiplication not commutative?
What is the difference between algebraic and geometric multiplicity?
How can I calculate the rank of a matrix?
When is a matrix diagonalizable?
What is the difference between a generating set and a basis?
For definitions and theorems, Zullo Linear Algebra Tutorbot will normally provide:
a precise mathematical statement;
an intuitive explanation;
a simple example;
a common mistake, misconception, or counterexample, when useful;
a reference to the relevant course material, when the reference can be verified.
You may also ask the Tutorbot to explain a concept in a different way.
For example:
Explain linear independence intuitively.
Explain the same concept using a geometric example.
Give me an example and a counterexample.
Explain this as if I were studying the topic for the first time.
Paste the complete exercise and indicate what kind of help you need.
Zullo Linear Algebra Tutorbot can provide:
guided help;
one hint only;
a complete solution;
verification of your own attempt.
By default, the Tutorbot uses a guided approach.
It identifies the relevant definitions or theorems and helps you complete the solution step by step.
It may first explain the strategy and then guide you through the calculations.
Determine whether the vectors
[
(1,0,1),\qquad (1,-1,0),\qquad (0,0,1)
]
form a basis of (\mathbb{R}^3).
The Tutorbot may guide you by asking:
How many vectors are given?
In which vector space do they belong?
Which criterion can be used to test whether they form a basis?
How can linear independence be checked?
This approach is recommended when you want to understand the method rather than immediately obtain the final answer.
Write:
/hint
followed by the complete exercise.
Zullo Linear Algebra Tutorbot will provide exactly one useful next step without revealing the complete solution.
/hint Determine whether the vectors
[
(1,2,0),\qquad (0,1,1),\qquad (1,3,1)
]
are linearly independent.
The Tutorbot may suggest setting up the equation
[
\alpha v_1+\beta v_2+\gamma v_3=0,
]
without carrying out all the subsequent calculations.
You can request another hint afterwards by writing:
Give me another hint.
The Tutorbot will then provide the next useful step.
Write:
/solution
followed by the complete exercise.
Zullo Linear Algebra Tutorbot will provide a complete, self-contained solution.
The solution will normally include:
the relevant definition or theorem;
the method to be used;
the important algebraic steps;
intermediate calculations;
a clear final conclusion.
/solution Solve the following system using Gaussian elimination:
[
\begin{cases}
x+2y-z=1,\
2x+y+z=4,\
3x+3y=5.
\end{cases}
]
When several methods are possible, you may specify the required one.
For example:
/solution Solve this system using Gaussian elimination, not Cramer’s rule.
/solution Compute the inverse using elementary row operations.
/solution Determine whether the matrix is diagonalizable using algebraic and geometric multiplicities.
You can paste your attempt and ask Zullo Linear Algebra Tutorbot to verify it.
Here is my solution. Please identify the first incorrect or unjustified step.
The Tutorbot will:
preserve the correct parts of your work;
identify the first error or missing justification;
explain why it is incorrect or incomplete;
show you how to repair the argument;
continue from the corrected step when appropriate.
It is better to submit your full reasoning rather than only the final answer.
My answer is (3). Is it correct?
I used Gaussian elimination and obtained the following row-echelon matrix:
[
\begin{pmatrix}
1&0&2&1\
0&1&-1&3\
0&0&0&1
\end{pmatrix}.
]
I concluded that the system has one solution. Is my conclusion correct?
The second request allows the Tutorbot to identify the precise conceptual error.
Write:
/oral
You may also specify one or more topics.
/oral
/oral Ask me about bases and dimension.
/oral Simulate an oral examination on eigenvalues and diagonalization.
/oral Test me on linear maps, kernel, and image.
Zullo Linear Algebra Tutorbot will:
ask one question at a time;
wait for your answer before continuing;
evaluate the correctness and precision of your answer;
identify missing hypotheses or inaccurate terminology;
ask a natural follow-up question;
focus on definitions, theorem statements, explanations, examples, and connections between topics;
occasionally ask you to perform a calculation or solve a short exercise, when appropriate.
The Tutorbot will not provide the ideal answer before you have attempted the question.
Tutorbot
Define a basis of a vector space.
Student
A basis is a set of vectors that generates the space.
Tutorbot
Your answer includes one of the two required properties. What additional condition must the generating set satisfy?
This method helps you practise giving precise and complete oral answers.
At the end of the simulation, the Tutorbot will provide a formative assessment containing:
topics understood well;
definitions stated correctly;
errors or omissions;
imprecise terminology;
topics and sections that should be revised.
Please note: This assessment is for practice only. It is not an official examination result or grade.
Write:
/written
You may also specify:
the expected duration;
the topics to include;
the number of exercises;
the desired difficulty.
/written
/written Create a 90-minute mock examination.
/written Include linear systems, vector spaces, and diagonalization.
/written Create four exercises at the level of the exercise sheets.
/written Create a difficult mock examination without solutions.
Zullo Linear Algebra Tutorbot will generate exercises consistent with the uploaded course material.
Solutions will not normally be provided immediately.
You should first:
attempt the exercises independently;
submit your work for verification;
request hints where needed;
request the complete solutions afterwards.
You can obtain the solutions by writing:
/solution Give me the solutions to the mock examination.
You may also submit one exercise at a time for correction.
Write:
/quiz
The Tutorbot will ask short questions one at a time.
It will adapt the difficulty according to your answers and explain mistakes before continuing.
/quiz Test me on matrices.
/quiz Test me on vector spaces and bases.
/quiz Test me on determinants and rank.
/quiz Test me on eigenvalues and diagonalization.
Quiz mode is particularly useful for checking:
definitions;
terminology;
theorem hypotheses;
short calculations;
conceptual differences;
common misconceptions.
Is every linearly independent set a basis?
After your answer, the Tutorbot will explain whether it is correct and, when necessary, provide an example or counterexample.
You can choose the difficulty level by writing:
/difficulty easy
/difficulty medium
/difficulty hard
The selected difficulty remains active during the conversation until you change it.
Suitable for reviewing basic concepts.
Questions may include:
definitions;
direct calculations;
simple examples;
basic matrix operations;
recognition of elementary properties.
Suitable for standard practice.
Questions are comparable to those in the course exercise sheets and may require several calculations or the application of a theorem.
Suitable for examination preparation.
Questions may require:
several connected steps;
a combination of different topics;
careful use of definitions and theorems;
interpretation of parameters;
comparison of different possible cases.
/difficulty hard
/written Create a mock examination on linear systems, vector spaces, and diagonalization.
You can change the difficulty at any time.
You can ask Zullo Linear Algebra Tutorbot to generate new exercises on a specific topic.
Give me three exercises on matrix multiplication.
Create five exercises on Gaussian elimination, ordered by difficulty.
Give me an exercise on bases of subspaces of (\mathbb{R}^4).
Create an exercise involving a parameter in a linear system.
Give me an exercise similar to one about diagonalization, but with different numbers.
You may also specify whether you want:
solutions;
hints;
final answers only;
full explanations;
common mistakes to avoid.
Create three medium-difficulty exercises on eigenvalues. Do not include the solutions yet.
Examples and counterexamples are essential for understanding linear algebra.
You can ask:
Give me two linearly independent vectors that are not orthogonal.
Give me three linearly dependent vectors in (\mathbb{R}^3), none of which is the zero vector.
Give me an invertible matrix that is not symmetric.
Give me a diagonalizable matrix with only two distinct eigenvalues.
Give me a matrix that is not diagonalizable.
Give me a generating set that is not a basis.
You can also ask the Tutorbot to explain why the example has the required properties.
Many difficulties in linear algebra come from confusing related concepts.
You can ask the Tutorbot to compare them directly.
Compare linear independence and orthogonality.
Compare a generating set and a basis.
Compare row equivalence and equality of matrices.
Compare eigenvalues and eigenvectors.
Compare algebraic and geometric multiplicity.
Compare homogeneous and non-homogeneous systems.
Compare kernel and image.
The Tutorbot will normally provide:
the definition of each concept;
the main difference;
an example;
a common misconception.
The quality of the answer depends on the quality of the request.
A good request should include:
the complete exercise or question;
all matrices, vectors, equations, parameters, and assumptions;
the method you are expected to use, when specified;
your own attempt, when available;
whether you want a hint, guidance, verification, or a complete solution.
Help me with matrices.
Check whether the matrix
[
A=
\begin{pmatrix}
1&2\
3&4
\end{pmatrix}
]
is invertible. Please explain the determinant criterion.
I do not understand bases.
Explain the difference between a generating set and a basis. Give one example of a generating set that is not a basis.
Solve this.
/hint Determine for which values of the parameter (a) the following system has a unique solution:
[
\begin{cases}
x+y+z=1,\
x+ay+z=2,\
x+y+az=3.
\end{cases}
]
You can write vectors using parentheses:
(v=(1,2,-1))
You can write a matrix in a simple text format:
(A=[[1,2],[3,4]])
or using mathematical notation:
[
A=
\begin{pmatrix}
1&2\
3&4
\end{pmatrix}.
]
For a linear system, include every equation clearly.
For example:
[
\begin{cases}
x+2y-z=1,\
2x-y+3z=4,\
x+y+z=0.
\end{cases}
]
When a parameter is involved, specify the set in which it varies.
For example:
Let (a\in\mathbb{R}). Determine the values of (a) for which the matrix is invertible.
According to the examination instructions provided for the course:
final examinations are reserved for students registered on the web-learning platform who possess a valid identity card;
the written examination is graded from (18/30) to (30/30);
written marks from (18/30) through (24/30) may be accepted without a compulsory theory oral examination;
an exercise may still be assigned during the oral phase;
the theory oral examination is recommended for students seeking a higher overall mark;
the written and oral components must be completed during the same examination session;
theory questions concern definitions, theorem statements, and explanations presented during lectures;
proofs and exercises are not required as part of the theory questions.
For administrative information that cannot be verified from the available course documents, consult the official course page or contact Prof. Zullo.
Zullo Linear Algebra Tutorbot must not invent:
examination dates;
deadlines;
room numbers;
grading rules;
office hours;
official course announcements.
Use Zullo Linear Algebra Tutorbot to:
understand mathematical methods;
practise independently;
check your reasoning;
identify gaps in your preparation;
revise definitions and theorems;
prepare more effectively for examinations.
The Tutorbot should support your learning, not replace it.
During an active examination or graded assessment, Zullo Linear Algebra Tutorbot must be used only in accordance with the instructor’s rules.
Generated work must not be presented as your own work.
Whenever possible, attempt an exercise independently before requesting the complete solution.
For each topic:
Review the relevant definition.
Explain the definition in your own words.
Identify all hypotheses and conditions.
Study one example.
Study one counterexample or common mistake.
Solve an exercise without assistance.
Use Zullo Linear Algebra Tutorbot to check your reasoning.
Correct your solution and repeat the exercise if necessary.
Practise the same topic in quiz or oral mode.
Record recurring mistakes and revise the relevant section of the course notes.
You can begin with one of the following requests.
Explain linear independence using an example and a counterexample.
Explain the Rouché–Capelli theorem and its hypotheses.
What is the difference between algebraic and geometric multiplicity?
/hint Help me determine whether these vectors form a basis.
/solution Solve this system using Gaussian elimination.
Check my calculation of the determinant and identify the first error.
/oral Ask me about vector spaces, subspaces, and bases.
/oral Simulate an oral examination on eigenvalues and diagonalization.
/written Create a 90-minute mock examination.
/difficulty hard
/written Create an examination on linear systems, rank, and diagonalization.
/quiz Test me on determinants and matrix rank.
/quiz Test me on bases and dimension.
Choose a topic, paste an exercise, or use one of the commands above.
Remember:
The goal is not only to obtain the correct answer, but to understand why the method works.