To determine if an equation represents a linear transformation, you need to check if it satisfies the two key properties of linearity: additivity and homogeneity of degree 1. Here are the steps and criteria:

1. Additivity (or Superposition Principle):

   • A transformation $T$ is additive if $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ for any vectors $\mathbf{u}$ and $\mathbf{v}$.

2. Homogeneity of Degree 1 (or Scalar Multiplication):

   • A transformation $T$ is homogeneous if $T(c\mathbf{u}) = cT(\mathbf{u})$ for any vector $\mathbf{u}$ and any scalar $c$.

To apply these criteria to an equation, follow these steps:

Steps to Identify a Linear Transformation:

1. Express the transformation in terms of matrices:

   • If the transformation can be written as $T(\mathbf{x}) = A\mathbf{x}$, where $A$ is a matrix and $\mathbf{x}$ is a vector, then $T$ is a linear transformation.

2. Check for Additivity:

   • Verify if $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$. Substitute $\mathbf{u} + \mathbf{v}$ into the transformation and see if it equals the sum of the transformations of $\mathbf{u}$ and $\mathbf{v}$.

3. Check for Homogeneity:

   • Verify if $T(c\mathbf{u}) = cT(\mathbf{u})$. Substitute $c\mathbf{u}$ into the transformation and see if it equals $c$ times the transformation of $\mathbf{u}$.

Example:

Consider the transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by: $T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x + 2y \\ -x + 4y \end{pmatrix}$

1. Matrix Representation: $T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$

   • This shows $T$ can be represented as a matrix multiplication.

2. Check Additivity: $T\begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} = \begin{pmatrix} 3(x_1 + x_2) + 2(y_1 + y_2) \\ -(x_1 + x_2) + 4(y_1 + y_2) \end{pmatrix}$ $= \begin{pmatrix} 3x_1 + 3x_2 + 2y_1 + 2y_2 \\ -x_1 - x_2 + 4y_1 + 4y_2 \end{pmatrix}$ $= \begin{pmatrix} 3x_1 + 2y_1 \\ -x_1 + 4y_1 \end{pmatrix} + \begin{pmatrix} 3x_2 + 2y_2 \\ -x_2 + 4y_2 \end{pmatrix}$ $= T\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + T\begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$

3. Check Homogeneity: $T\begin{pmatrix} c x \\ c y \end{pmatrix} = \begin{pmatrix} 3(c x) + 2(c y) \\ -(c x) + 4(c y) \end{pmatrix}$ $= \begin{pmatrix} c(3x) + c(2y) \\ c(-x) + c(4y) \end{pmatrix}$ $= c\begin{pmatrix} 3x + 2y \\ -x + 4y \end{pmatrix}$ $= c T\begin{pmatrix} x \\ y \end{pmatrix}$

Since both additivity and homogeneity are satisfied, $T$ is a linear transformation.

Summary

To identify if an equation represents a linear transformation:

1. Express it in matrix form if possible.
2. Check if it satisfies additivity.
3. Check if it satisfies homogeneity.

Would you like more details or have any questions?

Here are five questions you can ask:

1. How do I find the matrix representation of a linear transformation?
2. What are some examples of non-linear transformations?
3. Can you explain the properties of linear transformations in more detail?
4. How do linear transformations relate to vector spaces?
5. What is the significance of the kernel and image of a linear transformation?

Tip: When verifying linearity, always test with simple vectors and scalars to ensure the properties hold without extensive calculations.