How to Prove a Transformation is Linear: Step-by-Step Example in $\mathbb{R}^2$ | T(a_1,a_2)=(2a_1+a_2,a_1) | Example 1

Example

Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( T(a_1, a_2) = (2a_1 + a_2,\, a_1) \). Verify whether \( T \) is a linear transformation.

To show that \( T \) is linear, we must verify two fundamental properties: additivity and homogeneity.

Let \( \mathbf{u} = (x_1, y_1) \) and \( \mathbf{v} = (x_2, y_2) \) be arbitrary vectors in \( \mathbb{R}^2 \), and let \( c \in \mathbb{R} \) be any scalar.

Step 1: Check Additivity \( \big(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\big) \)

Compute the sum of vectors and its image under \( T \):

\[ \begin{aligned} \mathbf{u} + \mathbf{v} &= (x_1 + x_2,\, y_1 + y_2) \\[0.5em] T(\mathbf{u} + \mathbf{v}) &= \big(2(x_1 + x_2) + (y_1 + y_2),\, x_1 + x_2\big) \\ &= \big((2x_1 + y_1) + (2x_2 + y_2),\, x_1 + x_2\big) \end{aligned} \]

Now evaluate \( T(\mathbf{u}) + T(\mathbf{v}) \):

\[ \begin{aligned} T(\mathbf{u}) &= (2x_1 + y_1,\, x_1) \\ T(\mathbf{v}) &= (2x_2 + y_2,\, x_2) \\[0.5em] T(\mathbf{u}) + T(\mathbf{v}) &= \big((2x_1 + y_1) + (2x_2 + y_2),\, x_1 + x_2\big) \end{aligned} \]

Since \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \), additivity holds.

Step 2: Check Homogeneity \( \big(T(c\mathbf{u}) = c\,T(\mathbf{u})\big) \)

Scale vector \( \mathbf{u} \) and evaluate under \( T \):

\[ \begin{aligned} c\mathbf{u} &= (cx_1,\, cy_1) \\[0.5em] T(c\mathbf{u}) &= \big(2(cx_1) + cy_1,\, cx_1\big) \\ &= \big(c(2x_1 + y_1),\, cx_1\big) \end{aligned} \]

Now scale the image \( T(\mathbf{u}) \):

\[ c\,T(\mathbf{u}) = c(2x_1 + y_1,\, x_1) = \big(c(2x_1 + y_1),\, cx_1\big) \]

Since \( T(c\mathbf{u}) = c\,T(\mathbf{u}) \), homogeneity holds.

Since both properties are satisfied, \( T \) is a linear transformation. ■

Watch it worked out

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Engineering Mathematica · Linear Algebra Notes · MA25C01

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