Question

Which of the following forms a linear transformation:

• A. \( T: \mathbb{R}^2 \to \mathbb{R}, T(x, y) = xy \)
• B. \( T: \mathbb{R}^2 \to \mathbb{R}^3, T(x, y) = (x+1, 2y, x+y) \)
• C. \( T: \mathbb{R}^3 \to \mathbb{R}^2, T(x, y, z) = (|x|, 0) \)
• D. \( T: \mathbb{R}^2 \to \mathbb{R}^2, T(x, y) = (x+y, x) \)

Hint:

A transformation \(T\) from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is linear if and only if each component of the output vector is a linear combination of the input variables (e.g., \(T(x,y) = (ax+by, cx+dy)\)). Any constant terms (like the '+1' in option B), non-linear terms (\(xy\)), or functions like absolute value (\(|x|\)) will make the transformation non-linear.

Answer 1

The Correct Option is D

Step 1: Linear Transformation Definition:
A transformation \( T: V \to W \) is linear if it satisfies two conditions for all vectors \(\mathbf{u}, \mathbf{v} \in V\) and scalar \(c\):
1. Additivity: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. Homogeneity: \( T(c\mathbf{u}) = cT(\mathbf{u}) \)
A quick check is to see if \( T(\mathbf{0}) = \mathbf{0} \). If not, the transformation isn't linear.

Step 2: Transformation Analysis:
Let's analyze each transformation:
A. \( T(x, y) = xy \):
Checking homogeneity: Let \(\mathbf{u}=(x,y)\) and \(c\) be a scalar.
\( T(c\mathbf{u}) = T(cx, cy) = (cx)(cy) = c^2xy \).
\( cT(\mathbf{u}) = c(xy) = cxy \).
Since \( c^2xy eq cxy \) generally, this isn't linear.

B. \( T(x, y) = (x+1, 2y, x+y) \):
Checking the zero vector condition:
\( T(0,0) = (0+1, 2(0), 0+0) = (1, 0, 0) \).
Since \( T(\mathbf{0}) eq \mathbf{0} \), this isn't linear.

C. \( T(x, y, z) = (|x|, 0) \):
Checking homogeneity: Let \(\mathbf{u}=(x,y,z)\) and \(c=-1\).
\( T(-\mathbf{u}) = T(-x, -y, -z) = (|-x|, 0) = (|x|, 0) \).
\( -T(\mathbf{u}) = -( |x|, 0) = (-|x|, 0) \).
Since \( (|x|, 0) eq (-|x|, 0) \) for \(x eq 0\), this isn't linear.

D. \( T(x, y) = (x+y, x) \):
1. Additivity: Let \(\mathbf{u}=(x_1, y_1)\) and \(\mathbf{v}=(x_2, y_2)\).
\( T(\mathbf{u}+\mathbf{v}) = T(x_1+x_2, y_1+y_2) = ((x_1+x_2)+(y_1+y_2), x_1+x_2) \).
\( T(\mathbf{u})+T(\mathbf{v}) = (x_1+y_1, x_1) + (x_2+y_2, x_2) = (x_1+y_1+x_2+y_2, x_1+x_2) \).
Additivity holds.
2. Homogeneity: Let \(\mathbf{u}=(x,y)\) and \(c\) be a scalar.
\( T(c\mathbf{u}) = T(cx, cy) = (cx+cy, cx) = c(x+y, x) \).
\( cT(\mathbf{u}) = c(x+y, x) \).
Homogeneity holds.
Both conditions are met; thus, this is a linear transformation.

Step 3: Conclusion:
The transformation \( T(x, y) = (x+y, x) \) is the only linear one.