MMiDS 4.2: Self-Assessment Quiz

What is the rank of a matrix \( A \in \mathbb{R}^{n \times m} \)?

• a) The number of non-zero entries in \( A \).
• b) The dimension of the row space of \( A \).
• c) The dimension of the null space of \( A \).
• d) The trace of \( A \).

Which of the following is true about the rank of a matrix \( A \in \mathbb{R}^{n \times m} \)?

• a) \( \mathrm{rk}(A) \leq \min\{n,m\} \)
• b) \( \mathrm{rk}(A) \geq \max\{n,m\} \)
• c) \( \mathrm{rk}(A) = \mathrm{rk}(A^T) \) only if \( A \) is symmetric
• d) \( \mathrm{rk}(A) = \mathrm{rk}(A^T) \) only if \( A \) is square

Let \( A \in \mathbb{R}^{n \times k} \) and \( B \in \mathbb{R}^{k \times m} \). Which of the following is true?

• a) \( \mathrm{rk}(AB) \leq \mathrm{rk}(A) \)
• b) \( \mathrm{rk}(AB) \geq \mathrm{rk}(A) \)
• c) \( \mathrm{rk}(AB) = \mathrm{rk}(A) \)
• d) \( \mathrm{rk}(AB) = \mathrm{rk}(B) \)

Let \( A \in \mathbb{R}^{n \times m} \). What is the relationship between \( \mathrm{col}(A)^\perp \) and \( \mathrm{null}(A^T) \)?

• a) \( \mathrm{col}(A)^\perp = \mathrm{null}(A^T) \)
• b) \( \mathrm{col}(A)^\perp = \mathrm{null}(A) \)
• c) \( \mathrm{col}(A)^\perp = \mathrm{col}(A^T) \)
• d) \( \mathrm{col}(A)^\perp = \mathrm{row}(A^T) \)

What is the Rank-Nullity Theorem for a matrix \( A \in \mathbb{R}^{n \times m} \)?

• a) \( n = \mathrm{rk}(A) + \mathrm{dim}(\mathrm{null}(A)) \)
• b) \( m = \mathrm{rk}(A) + \mathrm{dim}(\mathrm{null}(A)) \)
• c) \( n = \mathrm{rk}(A) + \mathrm{dim}(\mathrm{null}(A^T)) \)
• d) Both b and c

Let \( A \in \mathbb{R}^{d \times d} \) and suppose \( \mathbf{v}_i \) and \( \mathbf{v}_j \) are eigenvectors corresponding to distinct eigenvalues \( \lambda_i \) and \( \lambda_j \). Which of the following is true?

• a) \( \mathbf{v}_i \) and \( \mathbf{v}_j \) are orthogonal
• b) \( \mathbf{v}_i \) and \( \mathbf{v}_j \) are linearly dependent
• c) \( \mathbf{v}_i \) and \( \mathbf{v}_j \) have the same Euclidean norm
• d) None of the above

Let \( A \in \mathbb{R}^{d \times d} \) be symmetric. Which of the following is true according to the Spectral Theorem?

• a) \( A \) has at most \( d \) distinct eigenvalues
• b) \( A \) has exactly \( d \) distinct eigenvalues
• c) \( A \) has at least \( d \) distinct eigenvalues
• d) The number of distinct eigenvalues of \( A \) is unrelated to \( d \)

Let \( A \in \mathbb{R}^{d \times d} \) be symmetric. Which of the following is equivalent to \( A \) being positive semidefinite?

• a) All eigenvalues of \( A \) are positive
• b) All eigenvalues of \( A \) are non-negative
• c) All eigenvalues of \( A \) are negative
• d) The trace of \( A \) is positive

Which of the following is true about the outer product of two vectors \( u \) and \( v \)?

• a) It is a scalar.
• b) It is a vector.
• c) It is a matrix of rank one.
• d) It is a matrix of rank zero.

If \( D \) and \( F \) are both square diagonal matrices, what is their product \( DF \)?

• a) A diagonal matrix whose diagonal entries are the sums of the corresponding diagonal entries of \( D \) and \( F \).
• b) A diagonal matrix whose diagonal entries are the products of the corresponding diagonal entries of \( D \) and \( F \).
• c) A matrix that is not necessarily diagonal.
• d) The identity matrix.