Verify the transformation 𝑇:𝑉2(𝑅)→𝑉2(𝑅) defined by 𝑇(𝑥,𝑦)= (3𝑥+2𝑦,3𝑥−4𝑦) is linear or not.

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.

Explore whether the transformation T(x, y) = (3x + 2y, 3x - 4y) is linear through detailed verification of additivity and homogeneity properties. Understand the concepts of linear transformations and their applications in vector spaces.

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