4) Find wheather the following krangformations are linear ar nat.

ⅰ> T: V₂ → ve defined by T (4,2)=(4+224)

11 R² defined by T (24/12, 13) = (14/10). T: R³ R² defined by 3 2

To determine whether the given transformations are linear, we need to check if they satisfy two key properties of linear transformations:

1. **Additivity**: For vectors \( \mathbf{u} \) and \( \mathbf{v} \) in the vector space, the transformation must satisfy \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
2. **Homogeneity (or scalar multiplication)**: For any vector \( \mathbf{v} \) in the vector space and any scalar \( c \), the transformation must satisfy \( T(c\mathbf{v}) = cT(\mathbf{v}) \).

Let's go through each transformation to check whether these conditions are satisfied.

### i) \( T: V_2 \rightarrow V_2 \) defined by \( T(4, 2) = (4 + 224) \)

The transformation here seems to be unclear or not fully specified. However, I assume it's trying to define a transformation for vectors in \( V_2 \). Let's re-interpret the transformation as:
- If \( T(4, 2) = (4 + 224) \), that would imply some operation on the vector \( (4, 2) \) to produce the result \(

4) Find whether the following transformations are linear or not.

ⅰ> T: V₂ → V₂ defined by T (4,2)=(4+224)

11) T: R² defined by T (24/12, 13) = (14/10). T: R³ → R² defined by (3, 2)

This problem explores how to determine if transformations between vector spaces are linear. Learn how to apply the additivity and homogeneity properties of linear transformations to test transformations like T: V₂ → V₂ and T: R² → R³. Suitable for undergraduate students studying linear algebra.

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