Quant Academy Math · Probability · Quant Finance
Phase 4 Exam

Linear Algebra - Phase Exam

20 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches
Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.
Q1 Easy
Which set is a subspace of \(\R^2\)?
The line \(y=x+1\)
The line \(y=2x\)
The unit circle
The first quadrant
Q2 Easy
For \(A:\R^5\to\R^3\) with \(\operatorname{rank}A=3\), the nullity is:
0
2
3
5
Q3 Easy
If \(\det A=0\) then:
\(A\) is invertible
\(A\) has 0 as an eigenvalue
\(A=0\)
\(A\) is symmetric
Q4 Easy
Similar matrices \(B=P^{-1}AP\) necessarily share:
The same entries
The same eigenvalues, trace, and determinant
The same eigenvectors
Nothing
Q5 Easy
The orthogonal projection of \(x\) onto \(U\) with orthonormal basis \((q_1,q_2)\) is:
\(x\)
\(\langle x,q_1\rangle q_1+\langle x,q_2\rangle q_2\)
\(q_1+q_2\)
\(\lVert x\rVert(q_1+q_2)\)
Q6 Easy
The spectral theorem guarantees that a real symmetric matrix has:
Complex eigenvalues
An orthonormal basis of eigenvectors and real eigenvalues
No eigenvectors
Distinct eigenvalues only
Q7 Easy
The singular values of \(A\) are:
The eigenvalues of \(A\)
The square roots of the eigenvalues of \(A^\top A\)
Always equal to 1
The diagonal entries of \(A\)
Q8 Medium
A list of 4 vectors in \(\R^3\) is:
Always independent
Always spanning
Necessarily dependent
Necessarily a basis
Q9 Medium
The system \(Ax=b\) has a solution if and only if:
\(b=0\)
\(A\) is square
\(b\in\operatorname{col}(A)\)
\(\det A\neq0\)
Q10 Medium
For \(A=\begin{bmatrix}2&1\\1&2\end{bmatrix}\), the eigenvalues are:
2 and 2
1 and 3
0 and 4
-1 and 3
Q11 Medium
An \(n\times n\) matrix is diagonalizable iff it has:
A nonzero determinant
n linearly independent eigenvectors
All positive eigenvalues
Distinct rows
Q12 Medium
The least-squares solution of \(Ax=b\) (independent columns) satisfies:
\(Ax=b\)
\(A^\top A\hat x=A^\top b\)
\(A\hat x=0\)
\(\hat x=A^\top b\)
Q13 Medium
A symmetric matrix has eigenvalues \(0\) and \(4\). It is:
Positive definite
Positive semidefinite but not definite
Indefinite
Negative definite
Q14 Medium
In PCA, the fraction of variance explained by the i-th principal component is:
\(\sigma_i\)
\(\lambda_i/\textstyle\sum_j\lambda_j\)
\(1/d\)
\(\lambda_i^2\)
Q15 Medium
Why is the dimension of a finite-dimensional space well-defined?
Because R^n is the only space
Because every basis has the same number of elements
Because bases are unique
Because vectors have finitely many components
Q16 Medium
A square matrix has a nontrivial null space. Then it is:
Invertible
Not invertible (singular)
Symmetric
Orthogonal
Q17 Medium
The product of all eigenvalues of a square matrix equals:
Its trace
Its determinant
Its rank
Always 1
Q18 Medium
The matrix \(\begin{bmatrix}1&1\\0&1\end{bmatrix}\) is:
Diagonalizable with distinct eigenvalues
Defective (not diagonalizable)
The identity
Singular
Q19 Medium
After Gram–Schmidt, the residual \(x-\operatorname{proj}_U x\) lies in:
\(U\)
\(U^\perp\)
the zero subspace only
the whole space with no structure
Q20 Hard
Why is every covariance matrix positive-semidefinite?
Its entries are positive
\(w^\top\Sigma w=\Var(w^\top X)\ge0\) for all \(w\)
It is diagonal
It is invertible
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