Steps to Verify Additivity and Homogeneity for Any Linear Map

📚 What is a Linear Map?

In mathematics, particularly linear algebra, a linear map (also called a linear transformation or linear function) is a mapping $T: V \rightarrow W$ between two vector spaces, $V$ and $W$, that preserves vector addition and scalar multiplication. These two properties, additivity and homogeneity, are the core requirements for a map to be considered linear.

📜 Historical Context

The concept of linear transformations has been central to mathematics since the development of linear algebra in the 19th century. Mathematicians like Arthur Cayley and Hermann Grassmann formalized these ideas, which have since become fundamental to various fields including physics, engineering, and computer science. Understanding linear maps allows us to represent and solve systems of linear equations, perform transformations in geometry, and analyze data in statistics.

🔑 Key Principles: Additivity and Homogeneity

A transformation $T: V \rightarrow W$ is linear if it satisfies the following two conditions:

• ➕ Additivity: For all vectors $\mathbf{u}$ and $\mathbf{v}$ in $V$, $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$.
• scale Homogeneity: For all vectors $\mathbf{u}$ in $V$ and all scalars $c$, $T(c\mathbf{u}) = cT(\mathbf{u})$.

✅ Steps to Verify Additivity and Homogeneity

Verifying that a given map $T$ is linear involves demonstrating that it satisfies both the additivity and homogeneity properties. Here's a step-by-step guide:

➕ Verifying Additivity

• 🎯 Step 1: Choose Arbitrary Vectors: Select two arbitrary vectors $\mathbf{u}$ and $\mathbf{v}$ from the domain vector space $V$. Define these vectors generally, for example, in $\mathbb{R}^2$, let $\mathbf{u} = (x_1, y_1)$ and $\mathbf{v} = (x_2, y_2)$.
• ➕ Step 2: Compute $T(\mathbf{u} + \mathbf{v})$: Find the expression for the sum of the vectors, $\mathbf{u} + \mathbf{v}$, and then apply the transformation $T$ to this sum.
• ➡️ Step 3: Compute $T(\mathbf{u}) + T(\mathbf{v})$: Find the expressions for $T(\mathbf{u})$ and $T(\mathbf{v})$ separately, and then add them together.
• ⚖️ Step 4: Compare: Verify that the expressions obtained in Step 2 and Step 3 are equal. If $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$, then $T$ satisfies the additivity property.

🔢 Verifying Homogeneity

• 💲Step 1: Choose an Arbitrary Vector and Scalar: Select an arbitrary vector $\mathbf{u}$ from the domain vector space $V$ and an arbitrary scalar $c$ (a real or complex number).
• 🧪 Step 2: Compute $T(c\mathbf{u})$: Find the expression for the scalar multiplication $c\mathbf{u}$, and then apply the transformation $T$ to this result.
• 💫 Step 3: Compute $cT(\mathbf{u})$: Find the expression for $T(\mathbf{u})$ and then multiply the result by the scalar $c$.
• 🔬 Step 4: Compare: Verify that the expressions obtained in Step 2 and Step 3 are equal. If $T(c\mathbf{u}) = cT(\mathbf{u})$, then $T$ satisfies the homogeneity property.

💡 Example 1: A Linear Transformation in $\mathbb{R}^2$

Consider the transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T(x, y) = (2x, x + y)$. Let's verify if it's linear.

Additivity:

Let $\mathbf{u} = (x_1, y_1)$ and $\mathbf{v} = (x_2, y_2)$.

• ➕ $T(\mathbf{u} + \mathbf{v}) = T(x_1 + x_2, y_1 + y_2) = (2(x_1 + x_2), (x_1 + x_2) + (y_1 + y_2)) = (2x_1 + 2x_2, x_1 + x_2 + y_1 + y_2)$.
• ➡️ $T(\mathbf{u}) + T(\mathbf{v}) = (2x_1, x_1 + y_1) + (2x_2, x_2 + y_2) = (2x_1 + 2x_2, x_1 + y_1 + x_2 + y_2)$.

Since $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$, additivity is satisfied.

Homogeneity:

Let $\mathbf{u} = (x, y)$ and $c$ be a scalar.

• 💲 $T(c\mathbf{u}) = T(cx, cy) = (2(cx), cx + cy) = (2cx, cx + cy)$.
• 💫 $cT(\mathbf{u}) = c(2x, x + y) = (2cx, cx + cy)$.

Since $T(c\mathbf{u}) = cT(\mathbf{u})$, homogeneity is satisfied.

Therefore, $T$ is a linear transformation.

🌍 Example 2: A Non-Linear Transformation in $\mathbb{R}^2$

Consider the transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T(x, y) = (x^2, y)$. Let's show that it is not linear.

Additivity:

Let $\mathbf{u} = (x_1, y_1)$ and $\mathbf{v} = (x_2, y_2)$.

• ➕ $T(\mathbf{u} + \mathbf{v}) = T(x_1 + x_2, y_1 + y_2) = ((x_1 + x_2)^2, y_1 + y_2) = (x_1^2 + 2x_1x_2 + x_2^2, y_1 + y_2)$.
• ➡️ $T(\mathbf{u}) + T(\mathbf{v}) = (x_1^2, y_1) + (x_2^2, y_2) = (x_1^2 + x_2^2, y_1 + y_2)$.

Since $T(\mathbf{u} + \mathbf{v}) \neq T(\mathbf{u}) + T(\mathbf{v})$ (because of the $2x_1x_2$ term), additivity is not satisfied. Therefore, $T$ is not a linear transformation.

📝 Conclusion

Verifying the linearity of a map is a fundamental skill in linear algebra. By systematically checking additivity and homogeneity, you can determine whether a given transformation preserves the essential properties of vector spaces. This knowledge is crucial for solving problems in various fields of science and engineering.