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A linear transformation is a special type of function between two vector spaces that ensures the principles of superposition apply. In simpler terms, you can think of it as a transformation that plays nice with addition and scalar multiplication. To determine if a function is a linear transformation, it must satisfy two main properties: **additivity** and **homogeneity**.

• Additivity refers to how the transformation handles the addition of vectors.
• Homogeneity relates to how the transformation deals with scalar multiplication.

For a function \( T \) to be a linear transformation, it needs these properties to align perfectly with its behavior. If even one is off, like in the given problem where additivity fails, the function isn't a linear transformation. Always remember: both conditions must hold true for it to be a verified linear transformation.

Additivity is one of the core properties required for a transformation to be considered linear. It reflects how the function interacts with the addition of two vectors. For a transformation \( T \), the function must satisfy the equation \( T(u+v) = T(u) + T(v) \). This means that if you add two vectors and then apply the transformation, it should equal adding the transformations of each vector separately.
Let's break it down further:

• You take two vectors, say \( u = (x_1, y_1) \) and \( v = (x_2, y_2) \).
• First, add the vectors together: \( u + v = (x_1 + x_2, y_1 + y_2) \).
• Apply the function: \( T(u + v) = (x_1 + x_2, 1) \).
• Then, separately, apply the function to each vector and add the results: \( T(u) = (x_1, 1) \) and \( T(v) = (x_2, 1) \), thus \( T(u) + T(v) = (x_1 + x_2, 2) \).

Now, if \( T(u+v) \) does not equal \( T(u) + T(v) \), then the function isn't maintaining the additivity property. In our example, the outputs were different, which is why the function could not be considered a linear transformation.

The homogeneity property is another essential check for linearity in transformations. It focuses on how the transformation interacts with scalar multiplication. In other words, if you multiply a vector by a scalar and then apply the transformation, it should be the same as scaling the transformed vector directly.
For homogeneity, a transformation \( T \) should satisfy: \( T(c \, \cdot \, u) = c \, \cdot \, T(u) \) where \( c \) is any scalar value and \( u \) is a vector.
Here's a step-by-step on what this implies:

• Consider a vector \( u = (x, y) \) and a scalar \( c \).
• Scale the vector: \( cu = (cx, cy) \).
• The transformation would be applied to get \( T(cu) = (cx, 1) \).
• Separately, transform the vector and then scale the transformation: \( T(u) = (x, 1) \); thus \( c \, \cdot \, T(u) = (cx, c) \).

The outputs of \( T(cu) \) and \( c \, \cdot \, T(u) \) must be identical for homogeneity to hold. In scenarios where the function doesn't meet this criterion, much like what happens with the given transformation's failure in additivity, the homogeneity check would similarly not pass, confirming the function isn't a linear transformation. Ensuring both the additivity and homogeneity properties are satisfied gives a complete check for linearity.