Question

In Exercises 1-5, a transformation \(T\) is given. Determine whether or not \(T\) is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+x_{2} \\ 3 x_{1}-x_{2} \end{array}\right] $$

Short answer

The transformation \( T \) is linear.

Step-by-step solution

Review the Definition of Linear Transformation

A transformation \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \) is linear if it satisfies two conditions for all vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^n \) and all scalars \( c \in \mathbb{R} \): 1) \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \); 2) \( T(c \mathbf{u}) = c T(\mathbf{u}) \).

Test Additivity Property

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 \( T(\mathbf{u} + \mathbf{v}) = \begin{bmatrix} (u_1 + v_1) + (u_2 + v_2) \ 3(u_1 + v_1) - (u_2 + v_2) \end{bmatrix} \). Simplifying, you get \( T(\mathbf{u} + \mathbf{v}) = \begin{bmatrix} (u_1 + u_2) + (v_1 + v_2) \ 3u_1 + 3v_1 - u_2 - v_2 \end{bmatrix} \).

Calculate \(T(\mathbf{u}) + T(\mathbf{v}) \)

Compute \( T(\mathbf{u}) = \begin{bmatrix} u_1 + u_2 \ 3u_1 - u_2 \end{bmatrix} \) and \( T(\mathbf{v}) = \begin{bmatrix} v_1 + v_2 \ 3v_1 - v_2 \end{bmatrix} \). So, \( T(\mathbf{u}) + T(\mathbf{v}) = \begin{bmatrix} (u_1 + u_2) + (v_1 + v_2) \ (3u_1 - u_2) + (3v_1 - v_2) \end{bmatrix} \). Simplifying yields \( \begin{bmatrix} (u_1 + u_2) + (v_1 + v_2) \ 3u_1 + 3v_1 - u_2 - v_2 \end{bmatrix} \).

Verify Additivity

From Step 2 and Step 3, we have \( T(\mathbf{u} + \mathbf{v}) = \begin{bmatrix} (u_1 + u_2) + (v_1 + v_2) \ 3u_1 + 3v_1 - u_2 - v_2 \end{bmatrix} \) and \( T(\mathbf{u}) + T(\mathbf{v}) = \begin{bmatrix} (u_1 + u_2) + (v_1 + v_2) \ 3u_1 + 3v_1 - u_2 - v_2 \end{bmatrix} \). These are equal, confirming the additivity property.

Test Homogeneity Property

Compute \( T(c\mathbf{u}) \), where \( c \mathbf{u} = \begin{bmatrix} cu_1 \ cu_2 \end{bmatrix} \). Thus, \( T(c \mathbf{u}) = \begin{bmatrix} cu_1 + cu_2 \ 3cu_1 - cu_2 \end{bmatrix} \).

Calculate \(cT(\mathbf{u})\)

From Step 3's result, \( cT(\mathbf{u}) = c \begin{bmatrix} u_1 + u_2 \ 3u_1 - u_2 \end{bmatrix} = \begin{bmatrix} c(u_1 + u_2) \ c(3u_1 - u_2) \end{bmatrix} = \begin{bmatrix} cu_1 + cu_2 \ 3cu_1 - cu_2 \end{bmatrix} \).

Verify Homogeneity

From Step 5 and Step 6, we have \( T(c \mathbf{u}) = \begin{bmatrix} cu_1 + cu_2 \ 3cu_1 - cu_2 \end{bmatrix} \) and \( cT(\mathbf{u}) = \begin{bmatrix} cu_1 + cu_2 \ 3cu_1 - cu_2 \end{bmatrix} \). These are equal, confirming the homogeneity property.

Conclusion

Since both additivity and homogeneity properties are satisfied, the transformation \( T \) is linear.

Key concepts

Additivity property

The additivity property is one of the key conditions for a transformation to be linear. It indicates that the transformation applied to a sum of vectors should yield the same result as summing the individual transformed vectors. This is a fundamental characteristic of linear transformations and can be written as:

• For vectors \( \mathbf{u} \) and \( \mathbf{v} \), \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).

In simple terms, when you add two vectors and then apply the transformation, the outcome should be the same as transforming both vectors separately and then adding the results.
This property ensures that the structure of vector addition is preserved under transformation.
If a transformation fails to satisfy this property, it is a strong indicator that it is not a linear transformation.
In our example, we've shown that the linearly transformed vectors and their sum equate, thereby confirming the additivity property.

Homogeneity property

The homogeneity property is the second essential trait defining a linear transformation. It states that if you scale a vector before applying the transformation, it should be equivalent to scaling the transformed vector. This condition can be expressed mathematically as:

• For a vector \(\mathbf{u}\) and scalar \(c\), \( T(c\mathbf{u}) = cT(\mathbf{u}) \).

In practical terms, this property indicates that the transformation respects scalar multiplication. This means multiplying a vector by any constant and then transforming it should be the same result as transforming the vector and then multiplying the result by that constant.
This property is crucial because it ensures the transformation maintains proportional relationships between vectors.
Our example demonstrates this with detailed calculations, confirming that the transformation \(T\) fulfills the homogeneity property, thereby supporting its linear nature.

Transformation matrix

In many cases, a linear transformation can be represented as a matrix. This is known as the transformation matrix. It acts as a concise and powerful tool for representing linear transformations, especially in higher dimensions. By using a matrix, every vector in the domain can be transformed with a single operation that greatly simplifies computations and visualization.

• A transformation matrix for \(T\) can often be derived directly from the transformation equation.
• For example, a transformation represented by matrix \(A\) applies to vector \(\mathbf{v}\) as: \(T(\mathbf{v}) = A\mathbf{v}\).

Understanding and utilizing the transformation matrix allows us to use matrix algebra to solve linear transformation problems efficiently and maximize computational capabilities.
For our transformation \(T\), the associated matrix can be extracted by observing the linear coefficients applied to each element of the input vector.

Matrix algebra

Matrix algebra forms the backbone of handling multiple calculations involving transformations, especially when dealing with multiple vectors or compound transformations. It not only supports operations like matrix addition and multiplication but also helps solve systems of equations conveniently. When applying transformations, using matrix algebra strengthens our computational strategies.

• Matrix multiplication corresponds to transforming vectors using a transformation matrix.
• It is important to remember the rules, such as the non-commutative property of matrix multiplication (i.e., \(AB eq BA\) necessarily), which influences how multiple transformations can be applied.

Using matrix algebra, we transform problems into matrix operations, which are extensively optimized in computational algorithms.
This approach is particularly useful for simulating transformations and solving larger systems, making it invaluable for both academic and real-world applications. In the context of our example, matrix algebra proved that \(T\) satisfies the linear transformation properties using these mathematical operations.