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?