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

Step 1: Concept:
We need to determine which maps $T : \mathbb{R}^2 \to \mathbb{R}^2$ satisfy the conditions of a linear transformation:
1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. $T(c\mathbf{u}) = c T(\mathbf{u})$
In particular, a necessary condition for linearity is $T(\mathbf{0}) = \mathbf{0}$.

Step 2: Key Formula or Approach:
A transformation $T(x_1, x_2)$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ is linear if and only if each component is a homogeneous linear combination of $x_1$ and $x_2$ (i.e., no constant terms, no powers, no absolute values).

Step 3: Step-by-step Explanation:

• Map A: $T(x_1, x_2) = (x_1, x_1 + x_2)$
Can be written in matrix form as: \[ T\begin{bmatrix} x_1 x_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 x_2 \end{bmatrix} \] Hence, $T$ is linear. Statement A is correct.

• Map B: $T(x_1, x_2) = (x_1 + x_2, x_2)$
Can be written in matrix form as: \[ T\begin{bmatrix} x_1 x_2 \end{bmatrix} = \begin{bmatrix} 1 & 1 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 x_2 \end{bmatrix} \] Hence, $T$ is linear. Statement B is correct.

• Map C: $T(x_1, x_2) = (x_1 + 1, x_2 + 2)$
Evaluating at the origin gives $T(0, 0) = (1, 2) \neq (0, 0)$.
Since $T(\mathbf{0}) \neq \mathbf{0}$, it is non-linear. Statement C is incorrect.

• Map D: $T(x_1, x_2) = (|x_1| - |x_2|, 0)$
Consider $c = -1$ and $\mathbf{u} = (1, 0)$: \[ T(-\mathbf{u}) = T(-1, 0) = (|-1| - |0|, 0) = (1, 0) \] \[ -T(\mathbf{u}) = -T(1, 0) = -(|1| - |0|, 0) = (-1, 0) \] Since $T(-\mathbf{u}) \neq -T(\mathbf{u})$, homogeneity fails. Statement D is incorrect.

• Map E: $T(x_1, x_2) = (x_1 - x_2, x_1 + x_2)$
Can be written in matrix form as: \[ T\begin{bmatrix} x_1 x_2 \end{bmatrix} = \begin{bmatrix} 1 & -1 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 x_2 \end{bmatrix} \] Hence, $T$ is linear. Statement E is correct.

Step 4: Final Answer:
Statements A, B, and E represent linear transformations. Therefore, option (D) is the correct answer.