e) T: R^3 -> R^3 is the transformation defined by T(x1,x2,x3) = (6 cube root x1x2x3, 6x1-4x2-x3, 8 cube root x3^2x1). A. T is a linear transformation. B. T does not preserve the zero vector. C. T does not preserve addition. D. T does not preserve scalar multiplication.

Let's analyze the transformation \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) defined by:

\[
T(x_1, x_2, x_3) = \left( 6 \sqrt[3]{x_1 x_2 x_3}, 6x_1 - 4x_2 - x_3, 8 \sqrt[3]{x_3^2 x_1} \right)
\]

We need to determine which of the following statements is true:
A. \( T \) is a linear transformation.  
B. \( T \) does not preserve the zero vector.  
C. \( T \) does not preserve addition.  
D. \( T \) does not preserve scalar multiplication.

### Step 1: Linear Transformation Check

For \( T \) to be a **linear transformation**, it must satisfy two properties:

1. **Preserve addition**:  
   \( T(\mathbf{v} + \mathbf{w}) = T(\mathbf{v}) + T(\mathbf{w}) \) for any vectors \( \mathbf{v}, \mathbf{w} \in \mathbb{R}^3 \).
   
2. **Preserve scalar multiplication**:  
   \( T(c \mathbf{v}) = c T(\mathbf{v}) \) for any scalar \( c \in \mathbb{R} \) and vector \( \mathbf{v} \in \mathbb{R}^3 \).

Let's check these properties for the given transformation.

### Step 2: Preservation of Addition
We need to check if \( T \) preserves vector addition, i.e., if for any vectors \( \mathbf{v} = (v_1, v_2, v_3) \) and \( \mathbf{w} = (w_1, w_2, w_3) \), the following holds:
\[
T(v_1 + w_1, v_2 + w_2, v_3 + w_3) = T(v_1, v_2, v_3) + T(w_1, w_2, w_3)
\]

However, from the first and third components of the transformation:
\[
T(v_1 + w_1, v_2 + w_2, v_3 + w_3) = \left( 6 \sqrt[3]{(v_1+w_1)(v_2+w_2)(v_3+w_3)}, \dots \right)
\]
This clearly does not equal:
\[
T(v_1, v_2, v_3) + T(w_1, w_2, w_3)
\]
because cube roots and products do not distribute over sums in this way. Hence, **T does not preserve addition**.

### Step 3: Preservation of Scalar Multiplication
We need to check if \( T \) preserves scalar multiplication, i.e., if for any scalar \( c \) and vector \( \mathbf{v} = (v_1, v_2, v_3) \), the following holds:
\[
T(c v_1, c v_2, c v_3) = c T(v_1, v_2, v_3)
\]

For the first component of \( T \):
\[
T(c v_1, c v_2, c v_3) = \left( 6 \sqrt[3]{(c v_1)(c v_2)(c v_3)}, \dots \right) = \left( 6 \sqrt[3]{c^3 v_1 v_2 v_3}, \dots \right) = \left( 6 c \sqrt[3]{v_1 v_2 v_3}, \dots \right)
\]
This is **not** equal to \( c T(v_1, v_2, v_3) \), as the second component does not multiply the scalar linearly, and the cube root affects the scalar \( c \) in the first and third components. Hence, **T does not preserve scalar multiplication**.

### Step 4: Zero Vector Check
For \( T \) to preserve the zero vector, we need to verify if \( T(0, 0, 0) = (0, 0, 0) \).

The transformation of the zero vector is:
\[
T(0, 0, 0) = \left( 6 \sqrt[3]{0 \cdot 0 \cdot 0}, 6 \cdot 0 - 4 \cdot 0 - 0, 8 \sqrt[3]{0^2 \cdot 0} \right) = (0, 0, 0)
\]
Thus, **T preserves the zero vector**.

### Conclusion:
- A. \( T \) is **not** a linear transformation.
- B. \( T \) **preserves** the zero vector.
- C. \( T \) **does not preserve addition**.
- D. \( T \) **does not preserve scalar multiplication**.

The correct answers are C and D.

Would you like more details on any of these concepts?

Here are 5 related questions to explore:
1. What are the properties of a linear transformation?
2. How does a non-linear transformation behave geometrically?
3. How can we test if a transformation preserves scalar multiplication in different dimensions?
4. What are some examples of linear and non-linear transformations in physics?
5. Why does preserving the zero vector matter in transformations?

**Tip**: Always test both addition and scalar multiplication to verify if a transformation is linear.

T: R^3 -> R^3 is the transformation defined by T(x1,x2,x3) = (6 cube root x1x2x3, 6x1-4x2-x3, 8 cube root x3^2x1). A. T is a linear transformation. B. T does not preserve the zero vector. C. T does not preserve addition. D. T does not preserve scalar multiplication.

Explore the analysis of a transformation T: R^3 -> R^3, defined by T(x1,x2,x3). This problem involves checking if the transformation preserves addition, scalar multiplication, and the zero vector, which are key properties of linear transformations. Ideal for students at the undergraduate level studying linear algebra.

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