Question

Which of the following functions $T : \mathbb{R}^2 \to \mathbb{R}^2$ are linear transformations? A. $T(x_1, x_2) = (x_1, x_1 + x_2)$
B. $T(x_1, x_2) = (x_1 + x_2, x_2)$
C. $T(x_1, x_2) = (x_1 + 1, x_2 + 2)$
D. $T(x_1, x_2) = (|x_1| - |x_2|, 0)$
E. $T(x_1, x_2) = (x_1 - x_2, x_1 + x_2)$
Choose the correct answer from the options given below:

• A. A, B, C Only
• B. B, C, E Only
• C. B, D, E Only
• D. A, B, E Only

Hint:

Quick check for linearity:
1. Non-zero $T(\mathbf{0}) \implies$ Non-linear.
2. Absolute values, products $x_1 x_2$, or powers $x_1^2 \implies$ Non-linear.

Answer 1

The Correct Option is D