Say that $v_1 = (1, 2, -2), v_2 = (3, 6, -4), v_3 = (0, 1, a^2-1)$ and $w_1 = (2, -a^2, -1), w_2 = (4, 6-2a, 2), w_3 = (0, 3, 2)$.
Part A asks for which values of a are $v_1, v_2, v_3$ linearly independent. I solved this by finding when the determinant of the matrix with $v_1, v_2, v_3$ is not equal to zero. The determinant is $-2$ for all values of $a$ so this is true for all $a \in \mathbb{R}$
Part B asks for which value of a there is at least 1 injective linear function $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $F(v_1) = w_1, F(v_2) = w_2, F(v_3) = w_3$. I don't have an immediate idea for this part of the question. I know that a function is injective if and only if the kernel is equal to zero, but I'm not sure what to do with that information.
Part C asks for which value of a is there a unique injective linear function $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $F(v_1) = w_1, F(v_2) = w_2, F(v_3) = w_3$. Again I am not sure what to do here.