To verify whether the transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T(x, y) = (3x + 2y, 3x - 4y)$ is linear, we need to check two properties:

1. Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. Homogeneity: $T(c\mathbf{u}) = cT(\mathbf{u})$ for any scalar $c$.

Step 1: Check Additivity

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

Calculate $T(\mathbf{u} + \mathbf{v})$:

$T(\mathbf{u} + \mathbf{v}) = T(x_1 + x_2, y_1 + y_2) = \left(3(x_1 + x_2) + 2(y_1 + y_2), 3(x_1 + x_2) - 4(y_1 + y_2)\right)$

Expanding this:

$T(\mathbf{u} + \mathbf{v}) = \left(3x_1 + 3x_2 + 2y_1 + 2y_2, 3x_1 + 3x_2 - 4y_1 - 4y_2\right)$

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

$T(\mathbf{u}) = T(x_1, y_1) = (3x_1 + 2y_1, 3x_1 - 4y_1)$

$T(\mathbf{v}) = T(x_2, y_2) = (3x_2 + 2y_2, 3x_2 - 4y_2)$

Adding these:

$T(\mathbf{u}) + T(\mathbf{v}) = \left(3x_1 + 2y_1, 3x_1 - 4y_1\right) + \left(3x_2 + 2y_2, 3x_2 - 4y_2\right)$

$T(\mathbf{u}) + T(\mathbf{v}) = \left(3x_1 + 3x_2 + 2y_1 + 2y_2, 3x_1 + 3x_2 - 4y_1 - 4y_2\right)$

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

Step 2: Check Homogeneity

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

Calculate $T(c\mathbf{u})$:

$T(c\mathbf{u}) = T(cx, cy) = (3(cx) + 2(cy), 3(cx) - 4(cy))$

Simplifying:

$T(c\mathbf{u}) = (c(3x + 2y), c(3x - 4y))$

Now calculate $cT(\mathbf{u})$:

$cT(\mathbf{u}) = c(3x + 2y, 3x - 4y) = (c(3x + 2y), c(3x - 4y))$

Since $T(c\mathbf{u}) = cT(\mathbf{u})$, the homogeneity property holds.

Conclusion

Since both additivity and homogeneity are satisfied, the transformation $T(x, y) = (3x + 2y, 3x - 4y)$ is linear.

Would you like to explore further details on this topic?

Related Questions:

1. What is the matrix representation of the linear transformation $T(x, y) = (3x + 2y, 3x - 4y)$?
2. Can you determine if the transformation $T(x, y) = (x^2, y^2)$ is linear?
3. How does the concept of linearity extend to transformations in higher dimensions, like $\mathbb{R}^3$?
4. What are some common applications of linear transformations in computer graphics?
5. How do you find the inverse of a linear transformation?

Tip:

In linear algebra, checking the properties of additivity and homogeneity is a fundamental way to determine whether a transformation is linear. Linear transformations preserve vector space structure, which is key in many areas of mathematics and physics.