Maths 2 Quiz 1: 30 January 2022, Set QPB1 (January 2022 term)

Consider the set $V$ together with the operations addition and scalar multiplication defined as follows:

$V = \{(x,y,z) \mid x,y,z \in \mathbb{R}\}$

Addition: $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2)$; $(x_1, y_1, z_1), (x_2, y_2, z_2) \in V$

Scalar multiplication:

$c(x,y,z) = \begin{cases} (0,0,0) & c = 0 \\ (cx, cy, \frac{z}{c}) & c \neq 0 \end{cases} \quad (x,y,z) \in V, \ c \in \mathbb{R}$

Consider the following statements.

1. There exists an element $0$ (called the zero vector of $V$) in $V$ such that $0 + v = v$, $\forall\, v \in V$.
2. For each $v$ in $V$, there exists an element $v'$, such that $v' + v = v + v' = 0$.
3. For each $v \in V$, $1v = v$.
4. For each $v \in V$ and for each pair $a, b \in \mathbb{R}$, $(a+b)v = av + bv$.
5. For each $a \in \mathbb{R}$ and for each pair $v_1, v_2 \in V$, $a(v_1 + v_2) = av_1 + av_2$.
6. For each $v \in V$ and for each pair $a, b \in \mathbb{R}$, $(ab)v = a(bv)$.

Choose the correct statements from the above.

Consider the following sets with the given operations defined as follows:

• $W_1 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, x + y + z = 1\}$

  Addition: $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2); \ (x_1, y_1, z_1), (x_2, y_2, z_2) \in W_1$

  Scalar multiplication: $c(x, y, z) = (cx, cy, cz); \ (x, y, z) \in W_1, \ c \in \mathbb{R}$

• $W_2 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, x = y - z\}$

  Addition: $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2); \ (x_1, y_1, z_1), (x_2, y_2, z_2) \in W_2$

  Scalar multiplication: $c(x, y, z) = (cx, cy, cz); \ (x, y, z) \in W_2, \ c \in \mathbb{R}$

Based on the above data, answer the given subquestions.

Which of the following options are true for W1?

Consider the following sets with the given operations defined as follows:

• $W_1 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, x + y + z = 1\}$

  Addition: $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2); \ (x_1, y_1, z_1), (x_2, y_2, z_2) \in W_1$

  Scalar multiplication: $c(x, y, z) = (cx, cy, cz); \ (x, y, z) \in W_1, \ c \in \mathbb{R}$

• $W_2 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, x = y - z\}$

  Addition: $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2); \ (x_1, y_1, z_1), (x_2, y_2, z_2) \in W_2$

  Scalar multiplication: $c(x, y, z) = (cx, cy, cz); \ (x, y, z) \in W_2, \ c \in \mathbb{R}$

Based on the above data, answer the given subquestions.

Which of the following options are true for W2?

Consider the vector space $\mathbb{R}^3$ with usual addition and scalar multiplication. Let $S$ be its subset defined as follows:

$S = \{(1, 0, 1), (0, 1, 1)\}$

and $W_1$ and $W_2$ be its vector subspaces defined as:

$W_1 = \{(x, y, z) \mid x = z, \text{ and } y = z; x, y, z \in \mathbb{R}\}$

$W_2 = \{(x, y, z) \mid z = x + y; x, y, z \in \mathbb{R}\}$

Answer the given subquestions.

Which of the following options is (are) true?

Consider the vector space $\mathbb{R}^3$ with usual addition and scalar multiplication. Let $S$ be its subset defined as follows:

$S = \{(1, 0, 1), (0, 1, 1)\}$

and $W_1$ and $W_2$ be its vector subspaces defined as:

$W_1 = \{(x, y, z) \mid x = z, \text{ and } y = z; x, y, z \in \mathbb{R}\}$

$W_2 = \{(x, y, z) \mid z = x + y; x, y, z \in \mathbb{R}\}$

Answer the given subquestions.

Which of the following options is (are) correct?

Consider the vector space $\mathbb{R}^3$ with usual addition and scalar multiplication. Let $S$ be its subset defined as follows:

$S = \{(1, 0, 1), (0, 1, 1)\}$

and $W_1$ and $W_2$ be its vector subspaces defined as:

$W_1 = \{(x, y, z) \mid x = z, \text{ and } y = z; x, y, z \in \mathbb{R}\}$

$W_2 = \{(x, y, z) \mid z = x + y; x, y, z \in \mathbb{R}\}$

Answer the given subquestions.

What is the dimension of W1?
Note: Enter your answer to the nearest integer.

Consider the vector space $\mathbb{R}^3$ with usual addition and scalar multiplication. Let $S$ be its subset defined as follows:

$S = \{(1, 0, 1), (0, 1, 1)\}$

and $W_1$ and $W_2$ be its vector subspaces defined as:

$W_1 = \{(x, y, z) \mid x = z, \text{ and } y = z; x, y, z \in \mathbb{R}\}$

$W_2 = \{(x, y, z) \mid z = x + y; x, y, z \in \mathbb{R}\}$

Answer the given subquestions.

What is the dimension of W2?
Note: Enter your answer to the nearest integer.

Suppose the following table represents the number of people (in thousands) travelling to the tourist spots from the metro cities in December 2019.

Due to covid these numbers got reduced in December 2020 which (in percentage of reduction with respect to December 2019) is given by:

i.e., the number of tourists from Chennai to Meghalaya in December 2020 is reduced to 25% of December 2019 (if 1000 were visiting in December 2019, only 250 are visiting in December 2020) and so on according to the table.

Using the given data answer the given subquestions.

If $A = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$ denotes the column vector representing the number of tourists to Manali (each element represent the number of tourists from different metro cities Chennai, Kolkata and Mumbai, respectively) in December 2019, and $C = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$ be the column vector representing the number of tourists to Manali in December 2020 (where $x_1$, $x_2$ and $x_3$ denote the number of tourists from different metro cities Chennai, Kolkata and Mumbai, respectively) and we have the following equation $BA = C$, then which of the following matrices can represent $B$?

Suppose the following table represents the number of people (in thousands) travelling to the tourist spots from the metro cities in December 2019.

Due to covid these numbers got reduced in December 2020 which (in percentage of reduction with respect to December 2019) is given by:

i.e., the number of tourists from Chennai to Meghalaya in December 2020 is reduced to 25% of December 2019 (if 1000 were visiting in December 2019, only 250 are visiting in December 2020) and so on according to the table.

Using the given data answer the given subquestions.

If $A = \begin{bmatrix} 3 & 2 & 1 \end{bmatrix}$ denotes the row vector representing the number of tourists from Kolkata (each element represent the number of tourists from Kolkata to Manali, Sikkim and Meghalaya, respectively) in December 2019, and $C = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}$ be the row vector representing the number of tourists from Kolkata in December 2020 (where $x_1$, $x_2$ and $x_3$ denote the number of tourists from Kolkata to Manali, Sikkim and Meghalaya, respectively) and we have the following equation $AB = C$, then which of the following matrices can represent $B$?

Suppose the following table represents the number of people (in thousands) travelling to the tourist spots from the metro cities in December 2019.

Due to covid these numbers got reduced in December 2020 which (in percentage of reduction with respect to December 2019) is given by:

i.e., the number of tourists from Chennai to Meghalaya in December 2020 is reduced to 25% of December 2019 (if 1000 were visiting in December 2019, only 250 are visiting in December 2020) and so on according to the table.

Using the given data answer the given subquestions.

Find the total number of tourists (in thousands) to Meghalaya in December 2020 from Chennai, Kolkata and Mumbai?
(Suppose your answer is 3000, then enter 3 as your answer)