Maths 2 Quiz 1: 16 October 2022 (September 2022 term)

Choose the set of correct options.

If addition and scalar multiplication on $V = \mathbb{R}^2$ is defined as follows:

Addition: $(x_1, y_1) + (x_2, y_2) = (0, 0)$; $(x_1, y_1), (x_2, y_2) \in V$

Scalar multiplication: $c(x, y) = (0, 0)$; $(x, y) \in V$, $c \in \mathbb{R}$

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 vector of $v \in V$ and for each pair $a, b \in \mathbb{R}, (a + b)v = av + bv$.
3. For each vector of $a \in \mathbb{R}$ and for each pair $v_1, v_2 \in V, a(v_1 + v_2) = av_1 + av_2$.
4. For each vector of $v \in V$ and for each pair $a, b \in \mathbb{R}, (ab)v = a(bv)$.

Which of the above statements is not true with respect to the addition and scalar multiplication on $V = \mathbb{R}^2$ defined above? (Enter the serial number of the statement which is not true. If statement 2 is incorrect, then enter 2 as your answer.)

Consider the following two statements:

P: $V = \mathbb{R}^2$, with the operations:

Addition:

$(x_1, y_1) + (x_2, y_2) = (x_1x_2, y_1y_2); \; (x_1, y_1), (x_2, y_2) \in V$

and

Scalar multiplication:

$c(x, y) = (cx, cy); \; (x, y) \in V, \; c \in \mathbb{R}$

is a vector space.

Q: Let $V$ be a vector space. If $u, v, w \in V$ are such that $au + bv + cw = 0$ for some scalars $a, b, c \in \mathbb{R}$ and $ac \neq 0$, then $\text{span}\{u, v\} = \text{span}\{v, w\}$.

Consider the following statements:

• Statement 1: P is true, but Q is false.
• Statement 2: Q is true, but P is false.
• Statement 3: Both P and Q are true.
• Statement 4: Both P and Q are false.

Which one of the above statements is correct? (e.g. if Statement 1 is correct, then enter 1 as your answer).

NOTE: Enter your answer to the nearest integer.

NOTE: Enter your answer to the nearest integer.

Based on the above data, answer the given subquestions.

Subset 1 is a subspace of dimension _______________. (Enter the numerical value only. Suppose the dimension is 3, then enter 3 as your answer.)

Based on the above data, answer the given subquestions.

Subset 2 is a subspace of dimension _______________. (Enter the numerical value only. Suppose the dimension is 3, then enter 3 as your answer.)

Based on the above data, answer the given subquestions.

Subset 3 is a subspace of dimension _______________. (Enter the numerical value only. Suppose the dimension is 3, then enter 3 as your answer.)

Based on the above data, answer the given subquestions.

Subset 4 is a subspace of dimension _______________. (Enter the numerical value only. Suppose the dimension is 3, then enter 3 as your answer.)

Suppose $W_1$ and $W_2$ are subspaces of $\mathbb{R}^3$ defined as follows:

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

and

$W_2 = \{(x, y, 0) \mid x, y \in \mathbb{R}\}$

with usual addition and scalar multiplication, i.e.,

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

Based on the above data, answer the given subquestions.

Suppose $W_1$ and $W_2$ are subspaces of $\mathbb{R}^3$ defined as follows:

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

and

$W_2 = \{(x, y, 0) \mid x, y \in \mathbb{R}\}$

with usual addition and scalar multiplication, i.e.,

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

Based on the above data, answer the given subquestions.

Suppose $W_1$ and $W_2$ are subspaces of $\mathbb{R}^3$ defined as follows:

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

and

$W_2 = \{(x, y, 0) \mid x, y \in \mathbb{R}\}$

with usual addition and scalar multiplication, i.e.,

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

Based on the above data, answer the given subquestions.

Which of the following options is true?

Suppose at $t = 2$ the distribution vector $X_2$ is $\begin{bmatrix} \frac{1}{3} \\ \frac{1}{2} \\ \frac{2}{3} \end{bmatrix}$. Which of the following options are true?

Choose the set of correct option(s).