1. Determine whether the following transformations are linear: Explain your
answer.
a. $F((x_1, x_2, x_3)^T) = (x_1 - x_2, x_2 - x_1)^T$
b. $F((x_1, x_2, x_3)^T) = (1, 2, x_1 + x_2 + x_3)^T$
c. $F((x_1)) = (x_1, 2x_1, 3x_1)^T$.
d. $F((x_1, x_2, x_3, x_4)^T) = (x_1, 0, 0, 0, x_2^2 + x_3^2 + x_4^2)^T$
2. Determine whether the following transformations are linear from $C([0, 1])$ to
$R$.
a. $L(f) = f(0)$, ($L := C([0, 1]) \to R$)
b. $L(f) = |f(0)|$, ($L := C([0, 1]) \to R$)
c. $L(f) = f'(0) + f(0)$. ($L := C^1([0, 1]) \to R$).
d. $L(f)(x) = x^2 + f(x)$, ($L := C([0, 1]) \to C([0, 1])$).
