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.