Maths 2 Quiz 1: 30 January 2022, Set QPB2 (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:

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

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:

• $W_1 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, 2x - y - z = 3\}$

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

• $W_2 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, x = 2y + 3z\}$

  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:

• $W_1 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, 2x - y - z = 3\}$

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

• $W_2 = \{(x, y, z) \mid x, y, z \in \mathbb{R}, x = 2y + 3z\}$

  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:

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

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

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

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

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:

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

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

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

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

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:

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

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

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

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

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:

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

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

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

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

Answer the given subquestions.

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

If $A = \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$ denotes the column vector representing the number of tourists to Sikkim (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 Sikkim 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 $B$?

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

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