练习题

If $f : R^{2} \to R$ , $f (x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2}$ then the initial guess for a gradient descent iteration is $x^{(0)} = (1, 1)^{T}$ and $α > 0$ , then $| f (x^{(1)}) | < | f (x^{(0)}) |$ if
1. $0 < α < 1$ ✅
2. $α > 0$
3. $α > \frac{1}{2}$
If $f : R^{2} \to R$ , $f (x_{1}, x_{2}) = e^{x_{1}} + x_{2}^{2}$ then the initial guess for a gradient descent iteration is $x^{(0)} = (0, 0)^{T}$ and $α > 0$ , then $| f (x^{(1)}) | < | f (x^{(0)}) |$ if
1. $α > \frac{1}{2}$
2. $α > 0$ ✅
3. $0 < α < 1$
If $f : R^{2} \to R$ , $f (x_{1}, x_{2}) = x_{1} e^{x_{2}}$ then the initial guess for a gradient descent iteration is $x^{(0)} = (1, 1)^{T}$ and $α = \frac{1}{2}$ , then :
1. $x^{(1)} = (1 - \frac{e}{2}, 1 - \frac{e}{2})^{T}$ ✅
2. $x^{(1)} = (1 + \frac{e}{2}, 1 + \frac{e}{2})^{T}$
3. $x^{(1)} = (\frac{1}{2} - \frac{e}{2}, \frac{1}{2} - \frac{e}{2})^{T}$
If $f : R^{2} \to R$ , $f (x_{1}, x_{2}) = e^{x_{1}} + x_{2}^{2}$ then the initial guess for a gradient descent iteration is $x^{(0)} = (0, 0)^{T}$ and $α > 0$ , then:
1. $x^{(1)} = (- α, 0)^{T}$ ✅
2. $x^{(1)} = (0, 0)^{T}$
3. $x^{(1)} = (- α, 2)^{T}$
For Standard IEEE, double precision representation is:
1. $F (2, 64, - 1024, 1023)$
2. none
3. $F (2, 53, - 1024, 1023)$ ✅
For Standard IEEE, single precision representation is:
1. $F (2, 24, - 128, 127)$ ✅
2. none
3. $F (2, 32, - 128, 127)$
Given two independent random variables X and Y, then
1. $p (x) = p (y)$
2. $p (y) = p (y | x)$ ✅
3. $p (x | y) = p (y)$
Given two random variables X and Y, Bayes Theorem implies that:
1. $p (x) = p (y) p (y | x) / p (y | x)$
2. $p (x) = p (x | y) p (y | x) / p (y)$
3. $p (y) = p (y | x) p (x) / p (x | y)$ ✅
If $f : R^{2} \to R$ , $f (x) = | | A x - b | |_{2}^{2}$ for $A \in R^{m * n}$ , $b \in R^{m}$ , then:
1. $\nabla f (x) = 2 A^{T} (A x - b)$ ✅
2. $\nabla f (x) = A^{T} (A x - b)$
3. $\nabla f (x) = A (A^{T} x - b)$
If $f : R^{2} \to R$ , $f (x) = | | A x - b | |_{2}^{2}$ for $A \in R^{m * n}$ , $b \in R^{m}$ , then minimum of x can be computed as:
1. $A^{T} A x = b$
2. $A^{T} A x = A^{T} x$
3. $A x = b$
4. $A^{T} A x = A^{T} b$ ✅
The machine precision $ϵ$ can be defined as:
1. The smallest number $ϵ$ such that $f l (1 + ϵ) = 1$
2. The smallest number $ϵ$ such that $f l (1 + ϵ) > 1$ ✅
3. none
Given two random variables X and Y such that $p (x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} x^{2}}$ and $p (y | x) = c e^{- | y - a x |}$ then the MAP reads:
1. $x * = a r g m i n_{x} | y - a x | + \frac{1}{2} x^{2}$ ✅
2. $x * = a r g m i n_{x} | y - a x |$
3. $x * = a r g m i n_{x} \frac{1}{2} (y - a x)^{2}$
Given two random variables X and Y such that $p (x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} x^{2}}$ and $p (y | x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} (y - a x)^{2}}$ then the MAP reads:
1. $x * = a r g m i n_{x} \frac{1}{2} (y - a x)^{2} + | x |$
2. $x * = a r g m i n_{x} \frac{1}{2} (y - a x)^{2} + \frac{1}{2} x^{2}$ ✅
3. $x * = a r g m i n_{x} \frac{1}{2} (y - a x)^{2}$
If $$ \begin{bmatrix} 2 & 1& -2 \ -1 & 0 & 1 \ -1 & 2 & 1 \end{bmatrix} $$
1. A is orthogonal
2. none ✅
3. A is symmetric and positive definite
If
$[\begin{matrix} 2 & 0 \\ 0 & 1 \end{matrix}]$
1. A is symmetric and positive definite ✅
2. A is non symmetric and not positive definite
3. A is symmetric but not positive definite
If
$[\begin{matrix} \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{- 2}{3} & \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{- 2}{3} \end{matrix}]$
1. A is symmetric and positive definite
2. A is non symmetric and not positive definite
3. A is orthogonal ✅
If
$[\begin{matrix} 2 & 2 & - 1 \\ 2 & 0 & 2 \\ - 1 & 2 & 3 \end{matrix}]$
1. A is symmetric but not definite positive ✅
2. A is symmetric and definite positive
3. A is orthogonal
If
$[\begin{matrix} - 1 & 1 \\ 0 & 3 \end{matrix}]$
1. A is symmetric but not definite positive
2. A is symmetric and definite positive
3. A non symmetric and not positive definite ✅
If
$[\begin{matrix} 9 & 6 \\ 6 & 5 \end{matrix}]$
1. A is symmetric but not definite positive
2. A is symmetric and definite positive ✅
3. A non symmetric and not positive definite
If
$[\begin{matrix} - 1 & 0 \\ 0 & 3 \end{matrix}]$
1. A is symmetric but not definite positive ✅
2. A is symmetric and definite positive
3. A non symmetric and not positive definite
If
$[\begin{matrix} 1 & 0 & 3 \\ 1 & 1 & 0 \\ - 1 & 1 & 2 \end{matrix}]$
1. rank(A)=2
2. rank(A)=1
3. rank(A)=3 ✅
If
$[\begin{matrix} 0 & 6 & 8 \\ 2 & 4 & 0 \\ 1 & 0 & 8 \end{matrix}]$
1. rank(A)=2
2. rank(A)=1
3. rank(A)=3 ✅
If
$[\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{matrix}]$
1. rank(A)=2 ✅
2. rank(A)=1
3. rank(A)=3
If
$[\begin{matrix} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$
1. rank(A)=3 ✅
2. rank(A)=4
3. rank(A)=2
If
$[\begin{matrix} 1 & - 1 & 1 \\ 2 & - 2 & 2 \\ - 1 & 1 & - 1 \end{matrix}]$
1. rank(A)=2
2. rank(A)=1 ✅
3. rank(A)=3
If
$[\begin{matrix} 0 & 2 & - 1 \\ 1 & 1 & 0 \\ 1 & 3 & - 1 \end{matrix}]$
1. rank(A)=2 ✅
2. rank(A)=1
3. rank(A)=3
If
$[\begin{matrix} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 4 \end{matrix}]$
1. rank(A)=3
2. rank(A)=4 ✅
3. rank(A)=2
If
$[\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 2 \end{matrix}]$
1. rank(A) = 1
2. rank(A) = 3
3. rank(A) = 2 ✅
Given two random variables X and such that $p (x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} x^{2}}$ and $p (y | x) = c e^{- | y - a x |}$ , then the MLE reads:
1. $x^{*} = a r g m i n_{x} | y - a x | + x^{2}$
2. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2}$
3. $x^{*} = a r g m i n_{x} | y - a x |$
Given two random variables X and such that $p (x) = c e^{- | x |}$ and $p (y | x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} (y - a x)^{2}}$ , then the MLE reads:
1. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2} + | x |$
2. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2} + \frac{1}{2} x^{2}$
3. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2}$
Given two random variables X and such that $p (x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} x^{2}}$ and $p (y | x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} (y - a x)^{2}}$ , then the MLE reads:
1. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2} + x^{2}$
2. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2} + \frac{1}{2} x^{2}$
3. $x^{*} = a r g m i n_{x} \frac{1}{2} (y - a x)^{2}$
If
$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$
1. The 2-norm of A is $| | A | |_{2} = 1$
2. The 2-norm of A is $| | A | |_{2} = 0$
3. The 2-norm of A is $| | A | |_{2} = 3$ ✅
if
$[\begin{matrix} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 4 \end{matrix}]$
1. The 2-norm of A is $| | A | |_{2} = 2$
2. The 2-norm of A is $| | A | |_{2} = 4$ ✅
3. The 2-norm of A is $| | A | |_{2} = 2$
If
$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$
1. The 2-norm of A is $| | A | |_{2} = 3$ ✅
2. The 2-norm of A is $| | A | |_{2} = 0$
3. The 2-norm of A is $| | A | |_{2} = 2$
If A is an $n \times n$ matrix, then
1. $| | A | |_{1} = \sqrt{\sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i, j}^{2}}$
2. $| | A | |_{1} = p (A^{T} A)$
3. None of the above ✅
If A is an $n \times n$ matrix, then
1. None of the above ✅
2. $| | A | |_{2} = p (A^{T} A)$
3. $| | A | |_{2} = \sqrt{\sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i, j}^{2}}$
If A is an $n \times n$ matrix, then
1. $| | A | |_{F} = \sqrt{\sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i, j}^{2}}$ ✅
2. None of the above
3. $| | A | |_{F} = p (A^{T} A)$
If $A \in R^{m \times n}, | | A | |_{p} = 0$ , then:
1. A=0 ✅
2. rank(A)=0
3. A can be both equal or not equal to 0
A matrix $A \in R^{n \times n}$ is orthogonal if:
1. $A^{- 1} A = I = A A^{- 1}$
2. $A^{T} A = I = A A^{T}$ ✅
3. $A = A^{T}$
If $X : Ω \to T$ is a continuous random variable, then a function $p : T \to R_{+}$ can be the PDF of X if:
1. $\int_{T} p (x) d x = 1$ ✅
2. $\int_{Ω} p (x) d x = 1$
3. $\int_{T} p (x) d x < \infty$
For a random variable $X : Ω \to T$ with $E [X] = 0$ , it holds:
1. $V a r (X) = E [X]$
2. $V a r (X) = E [X^{2}]$ ✅
3. $V a r (X) = 0$
If $X : Ω \to T$ is a discrete random variable, then a function $f_{X} : T \to [0, 1]$ can be the PDF of X if:
1. $\sum_{i \in Ω} f_{X} (i) = 1$
2. $\int_{Ω} f_{X} (x) d x = 1$
3. $\sum_{i \in T} f_{X} (i) = 1$ ✅
Given a discrete random Variable $X : Ω \to T$ with $T = 1, 2 \dots, 6$ and $f x = \frac{1}{6}, \frac{1}{6}, . . ., \frac{1}{6}$ then:
1. $E [X] = 21$
2. $E [X] = 3.5$ ✅
3. $E [X] = 1 / 6$
Given a continuous random Variable $X : Ω \to T$ with T=[0,1] and $p (x) = 3 x^{2}$ its PDF, then:
1. $E [X] = 2$
2. $E [X] = 3$
3. $E [X] = 3 / 4$ ✅
Given a continuous random Variable $X : Ω \to T$ with T=[0,1] and $p (x) = 2 x$ its PDF, then:
1. $E [X] = 2 / 3$ ✅
2. $E [X] = 2$
3. $E [X] = 1$
If $X : Ω \to T$ is a continuous random variable with PDF $p : T \to R_{+}$ , then:
1. $E [X] = \int_{T} p (x) d x$
2. $E [X] = \int_{Ω} x p (x) d x$
3. $E [X] = \int_{T} x p (x) d x$ ✅
f $X : Ω \to T$ is a continuous random variable, its PDF $p_{X} (x)$ is defined to be:
1. $P (X = x) = p_{X} (x)$
2. $P (X = x) = \int x p_{X} (x) d x$
3. $P (X = x) = \int_{A} p_{X} (x) d x$ ✅
f $X : Ω \to T$ is a continuous random variable, its PDF $p_{X} (x)$ is defined to be:
1. A function $p_{X} : T \to R_{+}$ ✅
2. A function $p_{X} : T \to [0, 1]$
3. A function $p_{X} : Ω \to [0, 1]$
If $X : Ω \to T$ is a discrete random variable, its PMF is:
1. $f_{x} (x) = \int P (x) d x$
2. $f_{x} (x) = P (X \in x)$
3. $f_{x} (x) = P (X = x)$ ✅
Given a discrete random Variable $X : Ω \to T$ with T={1,2,3} and $F x = \frac{1}{2}, \frac{1}{6}, \frac{1}{3}$ its PMF, then:
1. $E [X] = 6$
2. $E [X] = 2$
3. $E [X] = 11 / 6$ ✅
Given two discrete random variables $X_{1} : Ω \to T$ , $X_{2} : Ω \to T$ with T={1,2,3} and $f x_{1} = \frac{1}{3}, \frac{1}{3}, \frac{1}{3}$ and $f x_{2} = \frac{1}{2}, \frac{1}{6}, \frac{1}{3}$ then:
1. $E [X_{1}] > E [X_{2}]$ ✅
2. $E [X_{1}] = E [X_{2}]$
3. $E [X_{1}] < E [X_{2}]$
Given two discrete random variables $X_{1} : Ω \to T$ , $X_{2} : Ω \to T$ with T={1,2,3} and $f x_{1} = \frac{1}{2}, \frac{1}{6}, \frac{1}{3}$ and $f x_{2} = \frac{1}{6}, \frac{1}{3}, \frac{1}{2}$ then:
1. $E [X_{1}] > E [X_{2}]$
2. $E [X_{1}] = E [X_{2}]$
3. $E [X_{1}] < E [X_{2}]$ ✅
A random variable X is:
1. A function $X : Ω \to T$ ✅
2. A variable that returns random elements with known probability
3. A set that contains the possible outcomes of the experiment
If $Ω$ is the sample space, A is the event space and T is a subset of $R$ , a random variable X is:
1. A function $X : Ω \to A$
2. A function $X : A \to T$
3. A function $X : Ω \to T$ ✅
In normalized scientific notation and base $β = 10$ , if x = 2.71, then:
1. The mantissa of x is 0,271 and the exponential part is $10^{1}$
2. The mantissa of x is 2.71 and the exponential part is $10^{0}$ ✅
3. none
If $A = U Σ V^{T}$ is the SVD decomposition of an $m * n$ matrix, then a dyad $A_{i} = u_{i} v_{i}^{T}$ of A is:
1. None
2. A vector of length $m n$ that express some properties of A
3. A rank-1 matrix of dimension $m * n$ ✅
If $A = U Σ V^{T}$ is the SVD decomposition of an m*n matrix, then:
1. $A^{T} A = V^{T} Σ^{2} V$
2. $A^{T} A = U Σ^{2} U^{T}$
3. $A^{T} A = V Σ^{2} V^{T}$ ✅
If $A = U Σ V^{T}$ is the SVD decomposition of $A \in R^{m * n}$ , then its rank k approximation of $\tilde{A} (k)$ satisfies:
1. $\tilde{A} (k) = a r g m i n_{r k (B) = k} | | A - B | |_{2}$
2. $\tilde{A} (k) = a r g m i n_{r k (B) = k} | | A - B | |_{F}$ ✅
3. $\tilde{A} (k) = σ_{k + 1}$
If $A = U Σ V^{T}$ is the SVD decomposition of a $m * n$ matrix A, then:
1. The rows of $V^{T}$ are eigenvectors of $A A^{T}$
2. None
3. The columns of U are eigenvectors of $A A^{T}$ ✅
If $A = U Σ V^{T}$ is the SVD decomposition of a $m * n$ matrix A, then:
1. The rows of $V^{T}$ are eigenvectors of $A^{T} A$ ✅
2. None
3. The columns of U are eigenvectors of $A^{T} A$