How Do You Know If A Transformation Is Linear

Linear transformations are the backbone of linear algebra, acting as a bridge between different vector spaces while preserving their underlying structure. Understanding whether a transformation qualifies as linear is crucial for applying linear algebra concepts correctly. This article will guide you through the process of determining if a transformation is linear, complete with examples and explanations.

What is a Linear Transformation?

A transformation, often denoted as T, is a function that maps vectors from one vector space (V) to another vector space (W). This can be written as T: V → W. A linear transformation is a special type of transformation that adheres to two critical properties:

1. Additivity (Superposition): For any vectors u and v in V, the transformation of their sum must equal the sum of their individual transformations:

   T(u + v) = T(u) + T(v)

2. Homogeneity (Scaling): For any vector u in V and any scalar c, the transformation of the scalar multiple of the vector must equal the scalar multiple of the transformation of the vector:

   T(cu) = cT(u)

If a transformation satisfies both additivity and homogeneity, it is classified as a linear transformation. If it fails to satisfy either property, it is not linear.

The Two Key Properties Explained

Let's delve deeper into the meaning of additivity and homogeneity:

Additivity (Superposition)

Additivity ensures that the transformation preserves the structure of vector addition. In simpler terms, if you add two vectors before applying the transformation, the result should be the same as if you apply the transformation to each vector individually and then add the transformed vectors. This property is fundamental to maintaining the linearity of the transformation.

Homogeneity (Scaling)

Homogeneity guarantees that the transformation respects scalar multiplication. If you scale a vector by a constant before applying the transformation, the outcome should be the same as if you apply the transformation to the vector first and then scale the resulting transformed vector by the same constant. This property ensures that the relative magnitudes of vectors are preserved under the transformation.

Steps to Determine if a Transformation is Linear

To rigorously determine if a given transformation T is linear, follow these steps:

1. Choose Arbitrary Vectors: Select two arbitrary vectors, u and v, from the vector space V. These vectors should be general enough to represent any vector in the space. For example, in R², you might choose u = (x₁, y₁) and v = (x₂, y₂).

2. Verify Additivity:

   • Calculate T(u + v). This involves adding the vectors u and v and then applying the transformation T to the resulting vector.
   • Calculate T(u) + T(v). This involves applying the transformation T to each vector u and v individually and then adding the resulting transformed vectors.
   • Compare the results. If T(u + v) = T(u) + T(v) for all possible u and v, then the transformation satisfies the additivity property.

3. Verify Homogeneity:

   • Choose an arbitrary scalar c.
   • Calculate T(cu). This involves multiplying the vector u by the scalar c and then applying the transformation T to the resulting vector.
   • Calculate cT(u). This involves applying the transformation T to the vector u and then multiplying the resulting transformed vector by the scalar c.
   • Compare the results. If T(cu) = cT(u) for all possible u and all possible scalars c, then the transformation satisfies the homogeneity property.

4. Conclusion:

   • If the transformation satisfies both additivity and homogeneity, then it is a linear transformation.
   • If the transformation fails to satisfy either additivity or homogeneity, then it is not a linear transformation.

Examples of Linear and Non-Linear Transformations

Let's illustrate these steps with several examples:

Example 1: Linear Transformation in R²

Consider the transformation T: R² → R² defined by T(x, y) = (2x + y, x - y).

1. Additivity:

   • Let u = (x₁, y₁) and v = (x₂, y₂).
   • u + v = (x₁ + x₂, y₁ + y₂)
   • T(u + v) = T(x₁ + x₂, y₁ + y₂) = (2(x₁ + x₂) + (y₁ + y₂), (x₁ + x₂) - (y₁ + y₂)) = (2x₁ + 2x₂ + y₁ + y₂, x₁ + x₂ - y₁ - y₂)
   • T(u) = T(x₁, y₁) = (2x₁ + y₁, x₁ - y₁)
   • T(v) = T(x₂, y₂) = (2x₂ + y₂, x₂ - y₂)
   • T(u) + T(v) = (2x₁ + y₁ + 2x₂ + y₂, x₁ - y₁ + x₂ - y₂) = (2x₁ + 2x₂ + y₁ + y₂, x₁ + x₂ - y₁ - y₂)
   • Since T(u + v) = T(u) + T(v), additivity is satisfied.

2. Homogeneity:

   • Let u = (x, y) and c be a scalar.
   • cu = (cx, cy)
   • T(cu) = T(cx, cy) = (2(cx) + (cy), (cx) - (cy)) = (2cx + cy, cx - cy)
   • T(u) = T(x, y) = (2x + y, x - y)
   • cT(u) = c(2x + y, x - y) = (c(2x + y), c(x - y)) = (2cx + cy, cx - cy)
   • Since T(cu) = cT(u), homogeneity is satisfied.

Conclusion: Because T satisfies both additivity and homogeneity, it is a linear transformation.

Example 2: Non-Linear Transformation in R²

Consider the transformation T: R² → R² defined by T(x, y) = (x², y).

1. Additivity:
   • Let u = (x₁, y₁) and v = (x₂, y₂).
   • u + v = (x₁ + x₂, y₁ + y₂)
   • T(u + v) = T(x₁ + x₂, y₁ + y₂) = ((x₁ + x₂)², y₁ + y₂) = (x₁² + 2x₁x₂ + x₂², y₁ + y₂)
   • T(u) = T(x₁, y₁) = (x₁², y₁)
   • T(v) = T(x₂, y₂) = (x₂², y₂)
   • T(u) + T(v) = (x₁² + x₂², y₁ + y₂)
   • Since T(u + v) = (x₁² + 2x₁x₂ + x₂², y₁ + y₂) ≠ (x₁² + x₂², y₁ + y₂) = T(u) + T(v), additivity is not satisfied.

Conclusion: Because T fails to satisfy additivity, it is not a linear transformation. We don't even need to check homogeneity, as failing even one property disqualifies it.

Example 3: Another Non-Linear Transformation in R²

Consider the transformation T: R² → R² defined by T(x, y) = (x + 1, y).

1. Additivity:
   • Let u = (x₁, y₁) and v = (x₂, y₂).
   • u + v = (x₁ + x₂, y₁ + y₂)
   • T(u + v) = T(x₁ + x₂, y₁ + y₂) = (x₁ + x₂ + 1, y₁ + y₂)
   • T(u) = T(x₁, y₁) = (x₁ + 1, y₁)
   • T(v) = T(x₂, y₂) = (x₂ + 1, y₂)
   • T(u) + T(v) = (x₁ + 1 + x₂ + 1, y₁ + y₂) = (x₁ + x₂ + 2, y₁ + y₂)
   • Since T(u + v) = (x₁ + x₂ + 1, y₁ + y₂) ≠ (x₁ + x₂ + 2, y₁ + y₂) = T(u) + T(v), additivity is not satisfied.

Conclusion: Because T fails to satisfy additivity, it is not a linear transformation.

Example 4: A Linear Transformation from R³ to R²

Consider the transformation T: R³ → R² defined by T(x, y, z) = (x + y, y - z).

1. Additivity:

   • Let u = (x₁, y₁, z₁) and v = (x₂, y₂, z₂).
   • u + v = (x₁ + x₂, y₁ + y₂, z₁ + z₂)
   • T(u + v) = T(x₁ + x₂, y₁ + y₂, z₁ + z₂) = (x₁ + x₂ + y₁ + y₂, y₁ + y₂ - (z₁ + z₂)) = (x₁ + x₂ + y₁ + y₂, y₁ + y₂ - z₁ - z₂)
   • T(u) = T(x₁, y₁, z₁) = (x₁ + y₁, y₁ - z₁)
   • T(v) = T(x₂, y₂, z₂) = (x₂ + y₂, y₂ - z₂)
   • T(u) + T(v) = (x₁ + y₁ + x₂ + y₂, y₁ - z₁ + y₂ - z₂) = (x₁ + x₂ + y₁ + y₂, y₁ + y₂ - z₁ - z₂)
   • Since T(u + v) = T(u) + T(v), additivity is satisfied.

2. Homogeneity:

   • Let u = (x, y, z) and c be a scalar.
   • cu = (cx, cy, cz)
   • T(cu) = T(cx, cy, cz) = (cx + cy, cy - cz)
   • T(u) = T(x, y, z) = (x + y, y - z)
   • cT(u) = c(x + y, y - z) = (c(x + y), c(y - z)) = (cx + cy, cy - cz)
   • Since T(cu) = cT(u), homogeneity is satisfied.

Conclusion: Because T satisfies both additivity and homogeneity, it is a linear transformation.

Common Pitfalls and Tips

• Zero Vector: A linear transformation must map the zero vector in V to the zero vector in W. That is, T(0_V) = 0_W. If a transformation doesn't satisfy this, it's definitely not linear. This can be a quick initial check, but it's not sufficient to prove linearity. A transformation can map the zero vector to the zero vector and still be non-linear.
• Constants: Be wary of transformations that involve adding a constant term. These are often non-linear. The transformation T(x, y) = (x + 1, y) from Example 3 illustrates this.
• Non-Linear Operations: Transformations involving non-linear operations like squaring, square roots, trigonometric functions, or absolute values are generally non-linear. Example 2 (T(x, y) = (x², y)) demonstrates this.
• Superposition is Key: If you're unsure, always revert to the fundamental definitions of additivity and homogeneity. Work through the algebraic steps carefully.
• Counterexamples: To disprove linearity, you only need to find one specific example of vectors u, v, and a scalar c for which either additivity or homogeneity fails.

The Matrix Representation of Linear Transformations

A significant result in linear algebra is that every linear transformation between finite-dimensional vector spaces can be represented by a matrix. This provides a powerful tool for analyzing and computing linear transformations.

If T: V → W is a linear transformation, and V and W are finite-dimensional, then there exists a matrix A such that T(v) = Av for all v in V. The matrix A is called the matrix representation of T with respect to chosen bases for V and W.

Finding this matrix representation involves:

1. Choosing Bases: Select a basis for V and a basis for W. The standard basis is often the easiest choice, but other bases can be used.

2. Transforming Basis Vectors: Apply the transformation T to each basis vector in V.

3. Expressing as Linear Combinations: Express each transformed basis vector as a linear combination of the basis vectors in W. The coefficients of these linear combinations form the columns of the matrix A.

For example, let's revisit the linear transformation T: R² → R² defined by T(x, y) = (2x + y, x - y) from Example 1. Let's use the standard basis for R²: e₁ = (1, 0) and e₂ = (0, 1).

1. T(e₁) = T(1, 0) = (2(1) + 0, 1 - 0) = (2, 1)
2. T(e₂) = T(0, 1) = (2(0) + 1, 0 - 1) = (1, -1)

Since we're mapping to R², we express (2, 1) and (1, -1) in terms of the standard basis in R², which is trivial in this case. The coefficients become the columns of our matrix:

A = | 2 1 | | 1 -1 |

Therefore, T(v) = Av = | 2 1 | | x | = | 2x + y | | 1 -1 | | y | | x - y |

This confirms that the matrix A correctly represents the linear transformation T.

Why is Linearity Important?

Linear transformations are fundamental in many areas of mathematics, physics, engineering, and computer science because they:

• Preserve Structure: They maintain the underlying vector space structure, allowing us to apply linear algebra techniques.
• Simplify Problems: Many complex problems can be approximated or modeled using linear transformations, making them more tractable.
• Enable Analysis: They allow us to analyze the behavior of systems and predict their response to different inputs.
• Facilitate Computation: Matrix representations make it possible to perform linear transformations efficiently using computers.

Further Exploration

• Kernel and Image: The kernel (or null space) of a linear transformation T is the set of all vectors v in V such that T(v) = 0. The image (or range) of T is the set of all vectors w in W such that w = T(v) for some v in V.
• Isomorphisms: An isomorphism is a bijective (one-to-one and onto) linear transformation. Isomorphisms preserve all the essential algebraic properties of vector spaces.
• Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are crucial for understanding the behavior of linear transformations, particularly in the context of matrix diagonalization and solving systems of differential equations.

Conclusion

Determining whether a transformation is linear is a fundamental skill in linear algebra. By carefully verifying the additivity and homogeneity properties, you can confidently classify transformations and apply the appropriate linear algebra techniques. Remember to watch out for common pitfalls and use the matrix representation to simplify computations and analysis. Mastering this concept will unlock a deeper understanding of linear algebra and its applications in various fields.