Question 1

Let *A* be a 3 × 4 non-zero real matrix.\ Based on the above data, answer the given subquestions. The maximum value of *rank(A)* is \_\_\_\_\_\_\_\_\_\_\_\_\_ .\ **NOTE:** Enter your answer to the nearest integer. Let *A* be a 3 × 4 non-zero real matrix.\ Based on the above data, answer the given subquestions. The minimum 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: $$\langle ., . \rangle : V \times V \to \mathbb{R}$$ $$\langle (x_1, y_1), (x_2, y_2) \rangle = x_1x_2 + 4y_1y_2.$$ Which of the above conditions are satisfied for the above function?