Question

A transformation T is given. Determine whether or not T is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} x_{1}+1 \\ x_{2}+1 \end{array}\right] $$

Short answer

T is not linear because it fails the additivity test.

Step-by-step solution

Understand the Definition of a Linear Transformation

A transformation \( T \) is considered linear if it satisfies two main properties: Additivity and Homogeneity. 1. **Additivity**: For any vectors \( \mathbf{u} \) and \( \mathbf{v} \), the transformation should satisfy \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).2. **Homogeneity (Scalar Multiplication)**: For any vector \( \mathbf{u} \) and any scalar \( c \), the transformation should satisfy \( T(c \cdot \mathbf{u}) = c \cdot T(\mathbf{u}) \).

Check Additivity

Let \( \mathbf{u} = \begin{bmatrix} u_1 \ u_2 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix} \). Then, \( \mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \end{bmatrix} \).Calculate \( T(\mathbf{u} + \mathbf{v}) \):\[ T\left(\begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \end{bmatrix}\right) = \begin{bmatrix} (u_1 + v_1) + 1 \ (u_2 + v_2) + 1 \end{bmatrix} = \begin{bmatrix} u_1 + v_1 + 1 \ u_2 + v_2 + 1 \end{bmatrix} \]Now calculate \( T(\mathbf{u}) + T(\mathbf{v}) \):\[ T\left(\begin{bmatrix} u_1 \ u_2 \end{bmatrix}\right) + T\left(\begin{bmatrix} v_1 \ v_2 \end{bmatrix}\right) = \begin{bmatrix} u_1 + 1 \ u_2 + 1 \end{bmatrix} + \begin{bmatrix} v_1 + 1 \ v_2 + 1 \end{bmatrix} \]\[ = \begin{bmatrix} u_1 + v_1 + 2 \ u_2 + v_2 + 2 \end{bmatrix} \]Since \( T(\mathbf{u} + \mathbf{v}) eq T(\mathbf{u}) + T(\mathbf{v}) \), additivity is not satisfied.

Conclude that T is Not Linear

Since \( T \) fails to satisfy the additivity property, it is not a linear transformation. For a transformation to be linear, both additivity and homogeneity must hold true. Even without checking homogeneity, the failure of additivity alone disqualifies \( T \) from being a linear transformation.

Key concepts

Additivity

The concept of additivity is foundational to understanding linear transformations. Additivity requires that when two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), are transformed together by a linear transformation \( T \), the result should be the same as transforming each vector individually and then adding the results. In mathematical terms, this means \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).

In our example, we tested this property by considering two vectors \( \mathbf{u} = \begin{bmatrix} u_1 \ u_2 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix} \). After performing the necessary calculations, we found that \( T(\mathbf{u} + \mathbf{v}) \) did not equal \( T(\mathbf{u}) + T(\mathbf{v}) \).

The presence of \(+1\) added to each component of \( \mathbf{u} \) and \( \mathbf{v} \) in \( T \) disrupts the additivity. This means that the transformation \( T \) doesn't simply add up the vectors' influence; it adds an extra shift, violating the additivity requirement.

Homogeneity

Homogeneity is another crucial property required for a transformation to be linear. This property states that if you multiply a vector by a scalar and then transform it, the result should be the same as transforming the original vector and then multiplying the transformed result by that scalar.

In formal terms, for any vector \( \mathbf{u} \) and scalar \( c \), a linear transformation should satisfy \( T(c \cdot \mathbf{u}) = c \cdot T(\mathbf{u}) \).

Although we didn't specifically check homogeneity in our example due to the failure of additivity, it's important to understand how the presence of constants like those added in \( T \) can also disrupt homogeneity. Essentially, any operation on the vector that isn’t a multiplicative change can lead to a failure in meeting this criterion. Adding constants, such as \(+1\) to each vector component, means that any scalar multiplication will not simply "scale up" the transformation uniformly.

Vector Transformation

In the study of linear transformations, vector transformation is a key concept that involves changing vectors from one form to another while maintaining structural characteristics like direction and proportionality through certain operations.

Typically, linear transformations can be visualized as operations like rotating, scaling, or reflecting vectors. However, in our example, the transformation \( T \) does something more—it shifts each vector component by 1 unit.

• This is an example of an affine transformation, a broader category that includes shifts and translations in addition to linear transformations.
• Linear transformations strictly adhere to properties like additivity and homogeneity, allowing for operations to directly reflect changes in vector orientation and scale.

In cases like \( T \), where additional constants modify the vectors beyond these strict operations, the transformation doesn't fit into the linear category, affecting how the vectors behave in combination or under scaling. Understanding these distinctions is crucial in vector transformation tasks across various fields like computer graphics and physics.