If T : R" → R™ is a linear transformation, then T(0,) = 0m. O True O False Q5 Independence Every subset of a linearly independent set is linearly independent. O True O False
If T : R" → R™ is a linear transformation, then T(0,) = 0m. O True O False Q5 Independence Every subset of a linearly independent set is linearly independent. O True O False
Transcribed Image Text:### Linear Transformations and Independence
**Question 4:**
If \( T : \mathbb{R}^n \to \mathbb{R}^m \) is a linear transformation, then \( T(\theta_n) = \theta_m \).
- True
- False
**Question 5: Independence**
Every subset of a linearly independent set is linearly independent.
- True
- False
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**Explanation:**
- **Linear Transformations:** A function \( T \) is a linear transformation if it satisfies the properties of additivity and homogeneity. In the case where \( \theta_n \) and \( \theta_m \) are the zero vectors in \( \mathbb{R}^n \) and \( \mathbb{R}^m \) respectively, the statement \( T(\theta_n) = \theta_m \) must hold true because linear transformations map the zero vector to the zero vector.
- **Linearly Independent Sets:** A set of vectors is linearly independent if no vector in the set can be written as a combination of the other vectors. From this definition, any subset of a linearly independent set will also be linearly independent, as removing vectors cannot introduce a linear dependence.
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