What is dim range T =  1 in linear algebra

In linear algebra, the **dimension of the range** (or **image**) of a linear transformation \( T \), denoted as \( \dim(\text{range}(T)) \), refers to the number of linearly independent vectors in the range of \( T \). This is also called the **rank** of the transformation \( T \), which represents the number of distinct directions in the vector space that the transformation covers.

When you are given that \( \dim(\text{range}(T)) = 1 \), this means that the range (or image) of the transformation \( T \) is **one-dimensional**. In other words, the transformation maps all vectors to a subspace spanned by a single vector. All the output vectors are scalar multiples of that single vector, which indicates that the transformation essentially "collapses" the original space into a line.

This condition implies:
- The rank of \( T \) is 1.
- The image of \( T \) is a line through the origin in the codomain.
- \( T \) maps all input vectors to this one-dimensional subspace.

For example, if \( T: \mathbb{R}^n \to \mathbb{R}^m \), then all vectors in \( \mathbb{R}^m \) will lie along a single direction, and the transformation reduces the dimensionality of the space significantly.

Would you like to explore any details further?

Here are five related questions:
1. What is the nullity of a linear transformation, and how does it relate to the rank?
2. How is the Rank-Nullity Theorem used in this context?
3. Can a linear transformation with \( \dim(\text{range}(T)) = 1 \) still be injective or surjective?
4. What is the geometric interpretation of a linear transformation with rank 1?
5. How do you find the range of a matrix representing a linear transformation?

**Tip:** The Rank-Nullity Theorem states that the sum of the rank and the nullity of a linear transformation equals the dimension of the domain.

What is dim range T = 1 in linear algebra

Explore the concept of the dimension of the range of a linear transformation, with a focus on the case where dim(range(T)) = 1. Learn how this represents a rank-1 transformation, mapping vectors to a one-dimensional subspace, along with key concepts such as the Rank-Nullity Theorem. Ideal for undergraduate students.

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