MMiDS 1.2: Self-Assessment Quiz

Which of the following properties is satisfied by the inner product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \(\mathbb{R}^n\)?

• a) \(\langle \mathbf{u}, \mathbf{v} \rangle = \|\mathbf{u}\|_2 \|\mathbf{v}\|_2\)
• b) \(\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u}^T \mathbf{v}\)
• c) \(\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u} \cdot \mathbf{v} + 1\)
• d) \(\langle \mathbf{u}, \mathbf{v} \rangle = \|\mathbf{u} + \mathbf{v}\|_2^2\)

Let \(A \in \mathbb{R}^{n \times m}\) and \(B \in \mathbb{R}^{m \times p}\). What are the dimensions of the matrix product \(AB\)?

• a) \(n \times m\)
• b) \(m \times p\)
• c) \(n \times p\)
• d) \(p \times n\)

Which of the following is NOT a property of the transpose of a matrix?

• a) \((A^T)^T = A\)
• b) \((\gamma A)^T = \gamma A^T\) for any scalar \(\gamma \in \mathbb{R}\)
• c) \((A + B)^T = A^T + B^T\)
• d) \((AB)^T = A^T B^T\)

Let \(A \in \mathbb{R}^{n \times m}\) be a matrix and \(x \in \mathbb{R}^m\) be a column vector. Which of the following is true about the matrix-vector product \(Ax\)?

• a) \(Ax\) is a linear combination of the rows of \(A\) where the coefficients are the entries of \(x\)
• b) \(Ax\) is a linear combination of the columns of \(A\) where the coefficients are the entries of \(x\)
• c) \(Ax\) is a linear combination of the rows of \(A\) where the coefficients are the entries of \(x^T\)
• d) \(Ax\) is a linear combination of the columns of \(A\) where the coefficients are the entries of \(x^T\)

Let \(f: [a, b] \to \mathbb{R}\) be a continuous function whose derivative exists on \((a, b)\). According to the Mean Value Theorem, there exists \(c \in (a, b)\) such that:

• a) \(f(b) = f(a) + (b - a)f'(a)\)
• b) \(f(b) = f(a) + (b - a)f'(b)\)
• c) \(f(b) = f(a) + (b - a)f'(c)\)
• d) \(f(b) = f(a) + (b - c)f'(c)\)

Which of the following is true for a function \(f\) that is continuous on a closed interval \([a, b]\)?

• a) \(f\) attains its maximum value at \(x = a\).
• b) \(f\) does not attain a minimum value on \([a, b]\).
• c) \(f\) attains both a maximum and a minimum value on \([a, b]\).
• d) \(f\) attains its minimum value at \(x = b\).

Which of the following is true about the gradient \(\nabla f(x_0)\) of a continuously differentiable function \(f: \mathbb{R}^d \to \mathbb{R}\) at a local minimizer \(x_0\)?

• a) \(\nabla f(x_0) > 0\)
• b) \(\nabla f(x_0) < 0\)
• c) \(\nabla f(x_0) = 0\)
• d) \(\nabla f(x_0) \neq 0\)

Which of the following is NOT a property of the variance of a random variable?

• a) \(\mathrm{Var}[\alpha X + \beta] = \alpha^2 \mathrm{Var}[X]\) for \(\alpha, \beta \in \mathbb{R}\)
• b) \(\mathrm{Var}[X] = \E[(X - \E[X])^2]\)
• c) \(\mathrm{Var}[X] \ge 0\)
• d) \(\mathrm{Var}[X + Y] = \mathrm{Var}[X] + \mathrm{Var}[Y]\) for any random variables \(X\) and \(Y\)

If \(\mathbf{X}\) is a random vector in \(\mathbb{R}^d\) with mean vector \(\boldsymbol{\mu}_{\mathbf{X}}\) and covariance matrix \(\Sigma_{\mathbf{X}}\), which of the following expressions represents the covariance matrix \(\Sigma_{\mathbf{X}}\)?

• a) \(\E[(\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})^2]\)
• b) \(\E[(\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})(\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})^T]\)
• c) \(\E[(\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})^T (\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})]\)
• d) \(\E[(\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})^T] \E[(\mathbf{X} - \boldsymbol{\mu}_{\mathbf{X}})]\)

Which of the following is a property of the covariance matrix \(\Sigma_X\) of a random vector \(X\)?

• a) \(\Sigma_X\) is symmetric
• b) \(\Sigma_X\) is positive definite
• c) \(\Sigma_X\) is positive semidefinite
• d) Both a and c

Let \(X = (X_1, \ldots, X_d)\) be a standard Normal \(d\)-vector. Which of the following is true about its joint PDF \(f_X(x)\)?

• a) \(f_X(x) = \frac{1}{(2\pi)^{d/2}}\exp(-\frac{\|x\|^2}{2})\)
• b) \(f_X(x) = \frac{1}{(2\pi)^{d}}\exp(-\frac{\|x\|^2}{2})\)
• c) \(f_X(x) = \frac{1}{(2\pi)^{d/2}}\exp(-\|x\|^2)\)
• d) \(f_X(x) = \frac{1}{(2\pi)^{d}}\exp(-\|x\|^2)\)