Mathematics for Data Science II Quiz 2: 10 July 2022 (May 2022 term)

Choose the correct options.

An inner product on a vector space $V$ is a function $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ satisfying the following conditions:

Condition 1: $\langle v, v \rangle > 0$ for all $v \in V \setminus \{0\}$; $\langle v, v \rangle = 0$ if and only if $v = 0$.

Condition 2: $\langle v_1 + v_2, v_3 \rangle = \langle v_1, v_3 \rangle + \langle v_2, v_3 \rangle$.

Condition 3: $\langle v_1, v_2 \rangle = \langle v_2, v_1 \rangle$.

Condition 4: $\langle cv_1, v_2 \rangle = c\langle v_1, v_2 \rangle$

Let $V = \mathbb{R}^2$ and consider the function defined as:

$\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$

$\langle (x_1, x_2), (y_1, y_2) \rangle = x_1y_1 - x_1y_2 - x_2y_1 + x_2y_2.$

Which of the following are satisfied by the above function?

Let us consider the following matrices:

$A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}, B = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, C = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Consider the following pairs of matrices :

• Pair I: $A, B$
• Pair II: $A, C$
• Pair III: $B, C$

Choose the correct option from the following.

A norm on a vector space $V$ is a function

$\| \cdot \| : V \to \mathbb{R}$

$x \mapsto \|x\|$

satisfying the following conditions:

Condition 1: $\|x + y\| \leq \|x\| + \|y\|$ for all $x, y \in V$.

Condition 2: $\|cx\| = |c|\|x\|$ for all $c \in \mathbb{R}$ and for all $x \in V$.

Condition 3: $\|x\| \geq 0$ for all $x \in V$; $\|x\|=0$ if and only if $x = 0$.

Consider a function $\| \cdot \| : \mathbb{R}^3 \to \mathbb{R}$ defined as

$\|(x_1, x_2, x_3)\| = |x_1 + x_2 + x_3|$

on the vector space $\mathbb{R}^3$.

Which of the following are satisfied by the above function?

Let $T$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ defined as $T(x,y,z) = (x + y - z, y + z)$. Let $A$ be the matrix representation of $T$ with respect to the basis $\beta = \{(1,1,0), (0,1,1), (1,0,1)\}$ for the domain and the basis $\gamma = \{(1,1), (1,0)\}$ for the codomain.

$A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

Let $S = \{(x,y,z) \mid x = mz, y = nz; x,y,z \in \mathbb{R}\}$ be the nullspace of the $T$. Answer the subquestions based on the given data.

What is the value of d – a ?

Let $T$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ defined as $T(x,y,z) = (x + y - z, y + z)$. Let $A$ be the matrix representation of $T$ with respect to the basis $\beta = \{(1,1,0), (0,1,1), (1,0,1)\}$ for the domain and the basis $\gamma = \{(1,1), (1,0)\}$ for the codomain.

$A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

Let $S = \{(x,y,z) \mid x = mz, y = nz; x,y,z \in \mathbb{R}\}$ be the nullspace of the $T$. Answer the subquestions based on the given data.

What is the value of e – b?

Let $T$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ defined as $T(x,y,z) = (x + y - z, y + z)$. Let $A$ be the matrix representation of $T$ with respect to the basis $\beta = \{(1,1,0), (0,1,1), (1,0,1)\}$ for the domain and the basis $\gamma = \{(1,1), (1,0)\}$ for the codomain.

$A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

Let $S = \{(x,y,z) \mid x = mz, y = nz; x,y,z \in \mathbb{R}\}$ be the nullspace of the $T$. Answer the subquestions based on the given data.

What is the value of f – c?

Let $T$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ defined as $T(x,y,z) = (x + y - z, y + z)$. Let $A$ be the matrix representation of $T$ with respect to the basis $\beta = \{(1,1,0), (0,1,1), (1,0,1)\}$ for the domain and the basis $\gamma = \{(1,1), (1,0)\}$ for the codomain.

$A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

Let $S = \{(x,y,z) \mid x = mz, y = nz; x,y,z \in \mathbb{R}\}$ be the nullspace of the $T$. Answer the subquestions based on the given data.

What is the value of m?

Let $T$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ defined as $T(x,y,z) = (x + y - z, y + z)$. Let $A$ be the matrix representation of $T$ with respect to the basis $\beta = \{(1,1,0), (0,1,1), (1,0,1)\}$ for the domain and the basis $\gamma = \{(1,1), (1,0)\}$ for the codomain.

$A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

Let $S = \{(x,y,z) \mid x = mz, y = nz; x,y,z \in \mathbb{R}\}$ be the nullspace of the $T$. Answer the subquestions based on the given data.

What is the value of n?

Let $T$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ defined as $T(x,y,z) = (x + y - z, y + z)$. Let $A$ be the matrix representation of $T$ with respect to the basis $\beta = \{(1,1,0), (0,1,1), (1,0,1)\}$ for the domain and the basis $\gamma = \{(1,1), (1,0)\}$ for the codomain.

$A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

Let $S = \{(x,y,z) \mid x = mz, y = nz; x,y,z \in \mathbb{R}\}$ be the nullspace of the $T$. Answer the subquestions based on the given data.

Find out the nullity of T.

We apply the sequence of row operations on $A$, as follows:

• Step 1: $R_2 - 3R_1$
• Step 2: $-\frac{1}{3}R_2$
• Step 3: $R_1 - 2R_2$

Applying this row operations in the given order, the matrix $B$ is derived. Let

$B = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

What is the value of $a$ ?

We apply the sequence of row operations on $A$, as follows:

• Step 1: $R_2 - 3R_1$
• Step 2: $-\frac{1}{3}R_2$
• Step 3: $R_1 - 2R_2$

Applying this row operations in the given order, the matrix $B$ is derived. Let

$B = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

What is the value of $d$ ?

We apply the sequence of row operations on $A$, as follows:

• Step 1: $R_2 - 3R_1$
• Step 2: $-\frac{1}{3}R_2$
• Step 3: $R_1 - 2R_2$

Applying this row operations in the given order, the matrix $B$ is derived. Let

$B = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

What is the value of $c$ ?

If {(l, m, n)} is a basis of ker(T), then
Find the value of l if n is 1.

If {(l, m, n)} is a basis of ker(T), then
Find the value of m if n is 1.

Let $\beta = \{v_1, v_2\}$ be the orthonormal basis of the row space obtained by using the GramSchmidt process (with respect to usual inner product) applied on the ordered basis of the row space given by the first row and the second row of the matrix $A$. If

$v_2 = \frac{1}{\sqrt{195}}(b, c, d)$

What is the value of $\|30v_1\|$?

Let $\beta = \{v_1, v_2\}$ be the orthonormal basis of the row space obtained by using the GramSchmidt process (with respect to usual inner product) applied on the ordered basis of the row space given by the first row and the second row of the matrix $A$. If

$v_2 = \frac{1}{\sqrt{195}}(b, c, d)$

What is the value of $b$?

Let $\beta = \{v_1, v_2\}$ be the orthonormal basis of the row space obtained by using the GramSchmidt process (with respect to usual inner product) applied on the ordered basis of the row space given by the first row and the second row of the matrix $A$. If

$v_2 = \frac{1}{\sqrt{195}}(b, c, d)$

What is the value of $c$?

Let $\beta = \{v_1, v_2\}$ be the orthonormal basis of the row space obtained by using the GramSchmidt process (with respect to usual inner product) applied on the ordered basis of the row space given by the first row and the second row of the matrix $A$. If

$v_2 = \frac{1}{\sqrt{195}}(b, c, d)$

What is the value of $d$?