Maths 2 Quiz 1: 11 July 2021, Set POD21QZBQPA (May 2021 term)

Match the systems of linear equations in Column A with their number of solutions in column B and their geometric representation in Column C.

Consider the following two statements:

Statement 1: If $V$ is defined as follows:

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

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

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

Then $V$ is a vector space.

Statement 2:

If $W = Span\{(1,0,1), (0,1,1), (2,-1,1), (1,-2,-1)\}$, then $dim(W) = 3$.

Which of the following options is true?

Choose the set of correct options.

NOTE: Enter your answer to the nearest integer.

Anamika, Subhasis, and Shreya deposit $x, y,$ and $z$ amount of money (in thousands) respectively, every month in two different accounts $A_1$ and $A_2$. The amount of money is distributed in these two accounts $A_1$ and $A_2$ as $x - y$ and $2y + z$ respectively. This can be thought of as a linear transformation

$T : \mathbb{R}^3 \to \mathbb{R}^2$

defined by

$T(x, y, z) = (x - y, 2y + z) \quad .$

Note: A negative amount of money signifies the amount withdrawn from the accounts.

Using the above information, answer the given subquestions.

Which of the following vector spaces consists of vectors which could denote the amount of money deposited individually by Anamika, Subhasis and Shreya in a particular month such that in that month the amount deposited is 0 in both accounts A1 and A2.

Anamika, Subhasis, and Shreya deposit $x, y,$ and $z$ amount of money (in thousands) respectively, every month in two different accounts $A_1$ and $A_2$. The amount of money is distributed in these two accounts $A_1$ and $A_2$ as $x - y$ and $2y + z$ respectively. This can be thought of as a linear transformation

$T : \mathbb{R}^3 \to \mathbb{R}^2$

defined by

$T(x, y, z) = (x - y, 2y + z) \quad .$

Note: A negative amount of money signifies the amount withdrawn from the accounts.

Using the above information, answer the given subquestions.

Suppose at $t = 1$ the distribution vector $X_1$ is $\begin{bmatrix} 0 \\ 0 \\ \frac{1}{2} \end{bmatrix}$. Which of the following options is true?

Choose the set of correct options.