Free · 40 minutes · 20 questions · scored with a penalty

Practice test — Linear algebra.

20 questions drawn at random from the whole Linear algebra programme, mixed as in a real exam sitting: scoring +1/−0.25/0. At the end you receive by email the correct answer to every question and your weak points by topic — with an account you see them straight away on this page (the guided solution unlocks with your first lesson).

Each test mixes every topic of one first-year course at an Italian university, like a real exam sitting.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Topics covered by the test

Vector spaces and bases
Matrices and linear systems
Determinant and rank
Eigenvalues, eigenvectors, diagonalisation
Scalar product and quadratic forms

Two worked examples from the test

Question 1 · Algebra lineare · Vettori e spazi vettoriali

I vettori $v_{1} = (1, 2)$ e $v_{2} = (2, 4)$ in $R^{2}$ sono...

Alinearmente indipendenti
Blinearmente dipendenticorrect answer
Cortogonali
Duna base di $R^{2}$

Solution

Passo 1 — Verifico se uno è multiplo dell'altro. $v_{2} = (2, 4) = 2 \cdot (1, 2) = 2 v_{1} .$

Passo 2 — Conclusione. Esiste uno scalare $λ = 2$ tale che $v_{2} = λ v_{1}$ : i due vettori sono linearmente dipendenti (paralleli).

Risposta corretta: «linearmente dipendenti». Non possono essere una base di $R^{2}$ perché una base di uno spazio di dimensione 2 richiede 2 vettori indipendenti.

Question 2 · Algebra lineare · Vettori e spazi vettoriali

Per quale valore di $k$ i vettori $v_{1} = (1, 0, 1)$ , $v_{2} = (0, 1, 1)$ , $v_{3} = (1, 1, k)$ sono linearmente dipendenti?

A $k = 0$
B $k = 1$
C $k = 2$ correct answer
D $k = - 1$

Solution

Passo 1 — Impongo il determinante nullo. Tre vettori in $R^{3}$ sono dipendenti se il determinante della matrice che li ha per righe è zero.

$det 101011 11 k = 1 (k - 1) - 0 + 1 (0 - 1) = k - 1 - 1 = k - 2$

Passo 2 — Risolvo. $k - 2 = 0 \Rightarrow k = 2$ .

Verifica. Con $k = 2$ : $v_{3} = (1, 1, 2) = v_{1} + v_{2}$ , confermando la dipendenza.

Risposta corretta: $k = 2$ .

In the online practice test you will find 18 more questions like these; the correct answer to every question arrives by email.

What the Linear algebra exam is like

A written paper with a linear system to discuss as a parameter varies, a matrix to diagonalise, a basis to complete or orthonormalise and usually a quadratic form to classify; the oral asks for definitions with surgical precision and a few short proofs. It is the course where vocabulary counts as much as calculations.

Where the exam is lost

Confusing rank, dimension of the kernel and dimension of the image, and so discussing the parametric system badly. And declaring a matrix diagonalisable by looking only at the eigenvalues: for each eigenvalue the geometric multiplicity must equal the algebraic one, and that is exactly the check that gets skipped.

FAQ

How to read your score, in short.

What are the questions of the Linear algebra test like?

20 multiple-choice questions (4 options), drawn at random from a larger bank and spread over the WHOLE course programme together, not one topic at a time: vector spaces and bases, matrices and linear systems, determinant and rank, eigenvalues, eigenvectors, diagonalisation, scalar product and quadratic forms.

Is there a penalty for wrong answers?

Yes: +1 point for a correct answer, −0.25 for a wrong one, 0 for a question left blank — as in the TOLC (the Italian university admission tests). You should answer only when you are reasonably sure: it is the same logic as a real multiple-choice exam.

What does the final score mean?

15 out of 20 or more: you are ready, the programme is solid. 10-14: needs consolidating — review the topics of the questions you got wrong and try again. Under 10: to be reviewed calmly before you sit the test again.

Course still weak?

Let us look at your results together — the diagnostic is free — and I will tell you honestly what it takes to close the gap.

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