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) $⟨\mathbf{u},\mathbf{v}⟩=\Vert \mathbf{u}{\Vert }_2\Vert \mathbf{v}{\Vert }_2$
• b) $⟨\mathbf{u},\mathbf{v}⟩={\mathbf{u}}^T\mathbf{v}$
• c) $⟨\mathbf{u},\mathbf{v}⟩=\mathbf{u}\cdot \mathbf{v}+1$
• d) $⟨\mathbf{u},\mathbf{v}⟩=\Vert \mathbf{u}+\mathbf{v}{\Vert }_2^2$

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) $f(b)=f(a)+(b-a)f^{\prime }(a)$
• b) $f(b)=f(a)+(b-a)f^{\prime }(b)$
• c) $f(b)=f(a)+(b-a)f^{\prime }(c)$
• d) $f(b)=f(a)+(b-c)f^{\prime }(c)$

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 $\mathrm{\nabla }f(x_0)$ of a continuously differentiable function $f:{\mathbb{R}}^d\to \mathbb{R}$ at a local minimizer $x_0$?

• a) $\mathrm{\nabla }f(x_0)>0$
• b) $\mathrm{\nabla }f(x_0)<0$
• c) $\mathrm{\nabla }f(x_0)=0$
• d) $\mathrm{\nabla }f(x_0)\ne 0$

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) $\mathrm{Var}[X]=\text{E}[(X-\text{E}[X]{)}^2]$
• 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 ${\mathit{\mu }}_{\mathbf{X}}$ and covariance matrix ${\mathrm{\Sigma }}_{\mathbf{X}}$, which of the following expressions represents the covariance matrix ${\mathrm{\Sigma }}_{\mathbf{X}}$?

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

Which of the following is a property of the covariance matrix ${\mathrm{\Sigma }}_X$ of a random vector $X$?

• a) ${\mathrm{\Sigma }}_X$ is symmetric
• b) ${\mathrm{\Sigma }}_X$ is positive definite
• c) ${\mathrm{\Sigma }}_X$ is positive semidefinite
• d) Both a and c

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

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