Maths 2 Quiz 2: 13 March 2022, Set QPE1 (January 2022 term)

Let A be a 3 x 2 non-zero real matrix.
Based on the above data, answer the given subquestions.

The minimum value of rank(A) is _______________.
NOTE: Enter your answer to the nearest integer.

Let A be a 3 x 2 non-zero real matrix.
Based on the above data, answer the given subquestions.

The maximum value of nullity(A) is _______________.
NOTE: Enter your answer to the nearest integer.

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

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

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

$\langle (x_1, y_1), (x_2, y_2) \rangle = 2x_1x_2 + 3y_1y_2.$

Which of the above conditions are satisfied for the above function?

Determine whether the statements given in the subquestions are true or false.

If A or B is invertible, then AB and BA are similar matrices (i.e., AB is similar to BA).

Determine whether the statements given in the subquestions are true or false.

Any two scalar matrices are similar.

Determine whether the statements given in the subquestions are true or false.

If A is similar to B, then A^(k) is similar to B^(k), for any positive integer k.

Determine whether the statements given in the subquestions are true or false.

If A and B are two 3 × 3 matrices, which are similar to each other. Suppose the homogeneous system of linear equations Ax = 0 has a unique solution, then the homogeneous system of linear equations Bx = 0 also has a unique solution.

Anamika, Subhasis and Shreya pool together $x, y$, and $z$ amounts of money (in thousands) respectively, every month. The sum is distributed across three accounts $A_1$, $A_2$ and $A_3$ as $x + y + z$, $z - 2y$ and $2y - z$ respectively. This can be thought of as a linear transformation

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

defined by

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

Note: A negative amount of money signifies the amount withdrawn from the accounts. Answer the subquestions using the information given above.

Which of the following vector spaces consists of vectors which could denote the amount of money deposited by Anamika, Subhasis and Shreya in a particular month such that in that month the amount deposited is 0 in each of the accounts A1 , A2 and A3.

Anamika, Subhasis and Shreya pool together $x, y$, and $z$ amounts of money (in thousands) respectively, every month. The sum is distributed across three accounts $A_1$, $A_2$ and $A_3$ as $x + y + z$, $z - 2y$ and $2y - z$ respectively. This can be thought of as a linear transformation

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

defined by

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

Note: A negative amount of money signifies the amount withdrawn from the accounts. Answer the subquestions using the information given above.

Find out nullity(T).
NOTE: Enter your answer to the nearest integer.

Anamika, Subhasis and Shreya pool together $x, y$, and $z$ amounts of money (in thousands) respectively, every month. The sum is distributed across three accounts $A_1$, $A_2$ and $A_3$ as $x + y + z$, $z - 2y$ and $2y - z$ respectively. This can be thought of as a linear transformation

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

defined by

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

Note: A negative amount of money signifies the amount withdrawn from the accounts. Answer the subquestions using the information given above.

Find out rank(T).
NOTE: Enter your answer to the nearest integer.

Anamika, Subhasis and Shreya pool together $x, y$, and $z$ amounts of money (in thousands) respectively, every month. The sum is distributed across three accounts $A_1$, $A_2$ and $A_3$ as $x + y + z$, $z - 2y$ and $2y - z$ respectively. This can be thought of as a linear transformation

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

defined by

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

Note: A negative amount of money signifies the amount withdrawn from the accounts. Answer the subquestions using the information given above.

Which of the following options is true?

Anamika, Subhasis and Shreya pool together $x, y$, and $z$ amounts of money (in thousands) respectively, every month. The sum is distributed across three accounts $A_1$, $A_2$ and $A_3$ as $x + y + z$, $z - 2y$ and $2y - z$ respectively. This can be thought of as a linear transformation

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

defined by

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

Note: A negative amount of money signifies the amount withdrawn from the accounts. Answer the subquestions using the information given above.