Question

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

Short answer

The transformation \( T \) is not linear because it fails both additivity and scalar multiplication properties.

Step-by-step solution

Definition of Linearity

To determine if the transformation \( T \) is linear, we need to verify if it satisfies two conditions for all vectors \( \mathbf{u} \) and \( \mathbf{v} \) and all scalars \( c \): 1) \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \) (Additivity), and 2) \( T(c\mathbf{u}) = cT(\mathbf{u}) \) (Scalar Multiplication).

Check Additivity

Consider \( \mathbf{u} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \). Calculate \( T(\mathbf{u}+\mathbf{v}) = T\left(\begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix}\right) \) which becomes \( \begin{bmatrix} (x_1+y_1) + (x_2+y_2)^2 \ (x_1+y_1) - (x_2+y_2) \end{bmatrix} \). Meanwhile, calculate \( T(\mathbf{u}) + T(\mathbf{v}) = \begin{bmatrix} x_1+x_2^2 \ x_1-x_2 \end{bmatrix} + \begin{bmatrix} y_1+y_2^2 \ y_1-y_2 \end{bmatrix} = \begin{bmatrix} x_1 + y_1 + x_2^2 + y_2^2 \ x_1 + y_1 - x_2 - y_2 \end{bmatrix} \). Noticing the discrepancy of \( (x_2+y_2)^2 eq x_2^2 + y_2^2 \), additivity fails.

Check Scalar Multiplication

Consider \( c \cdot \mathbf{u} = \begin{bmatrix} cx_1 \ cx_2 \end{bmatrix} \). Calculate \( T(c\mathbf{u}) = T\left(\begin{bmatrix} cx_1 \ cx_2 \end{bmatrix}\right) = \begin{bmatrix} cx_1 + (cx_2)^2 \ cx_1 - cx_2 \end{bmatrix} \). However, \( cT(\mathbf{u}) = c\begin{bmatrix} x_1+x_2^2 \ x_1-x_2 \end{bmatrix} = \begin{bmatrix} c(x_1 + x_2^2) \ c(x_1-x_2) \end{bmatrix} = \begin{bmatrix} cx_1 + cx_2^2 \ cx_1 - cx_2 \end{bmatrix} \). Since \( (cx_2)^2 eq cx_2^2 \), scalar multiplication fails.

Conclusion

Since neither additivity nor scalar multiplication properties are satisfied, the transformation \( T \) is not linear.

Key concepts

Additivity

When determining if a transformation is linear, one crucial aspect to check is **additivity**. In simple terms, additivity means that if you apply the transformation to the sum of two vectors, the result should equal the sum of the transformations applied to each vector individually. This is mathematically expressed as:

• \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)

For the given transformation problem, this was tested, but the results showed a discrepancy.
Instead of both sides being equal, there was a mismatch: \((x_2 + y_2)^2\) did not equal \(x_2^2 + y_2^2\). Because of this flaw, the additivity property does not hold.

Understanding additivity is essential in determining linearity. When this condition fails, as shown in the exercise, it indicates that the transformation isn't fully linear.

Scalar Multiplication

Another vital concept in examining the linearity of a transformation is **scalar multiplication**. This involves testing whether applying a transformation to a scaled (multiplied by a constant) vector yields the same result as scaling the transformation of the vector. Mathematically, this is represented as:

• \( T(c\mathbf{u}) = cT(\mathbf{u}) \)

In the exercise, this property was tested by comparing the transformed scaled vector to the scaled transformation of the vector.
However, it was found that scaling and then squaring a component, \((cx_2)^2\), did not match the scaling of its square, \(cx_2^2\).
This mismatch led to the conclusion that the scalar multiplication property does not hold for this transformation.

Scalar multiplication is crucial for a transformation to be considered linear, and failure in this property further confirms the non-linearity of the transformation in question.

Transformation Matrix

In linear algebra, the concept of a **transformation matrix** is pivotal for understanding linear transformations. A transformation matrix gives a concise way to describe these transformations and to understand their effects on vectors. For a transformation to be linear, it can usually be represented by a matrix multiplication.
Every linear transformation from one vector space to another can be expressed using a transformation matrix \( A \) such that:

• \( T(\mathbf{x}) = A\mathbf{x} \)

In the exercise example, a transformation matrix cannot be easily defined due to the presence of non-linear elements such as \(x_2^2\).
The absence of a simple matrix representation is another indicator that the transformation does not meet the criteria for linearity. Being able to express a transformation as a matrix is not just a formality but a vital confirmation of its linear nature.

Non-linearity

**Non-linearity** in a transformation means it does not obey the rules of additivity and scalar multiplication, both critical for linear transformations. When a transformation is non-linear, it includes one or more elements or operations that disrupt these linear properties.
In the exercise provided, the term \(x_2^2\) introduces non-linearity. Such a term is problematic because squaring a component is inherently a non-linear operation.

• This disrupts additivity because when components are squared independently, they do not simply add up linearly.
• It breaks scalar multiplication as scaling before squaring yields different results from squaring and then scaling.

Understanding non-linearity helps us identify why certain transformations cannot be expressed in a simple, linear framework. Recognizing these deviations is key in distinguishing complex transformations from those easily managed within linear algebra's defined parameters.