The problem provided is about a matrix transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ that is one-to-one (injective). We need to determine the relationship between $m$ and $n$.

Explanation:

A matrix transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be represented by an $m \times n$ matrix. For the transformation to be one-to-one, its kernel must only contain the zero vector. This implies that the columns of the matrix must be linearly independent.

If the matrix has $n$ columns (since it maps from $\mathbb{R}^n$), and for the columns to be linearly independent, we must have $n \leq m$. This is because there cannot be more linearly independent vectors in $\mathbb{R}^m$ than the dimension $m$ itself.

Conclusion:

The correct answer is:

B. $n \leq m$

Would you like more details on this topic, or do you have any other questions?

Related Questions:

1. What is the difference between injective and surjective transformations?
2. How can you determine if a matrix transformation is onto (surjective)?
3. What is the rank of a matrix, and how does it relate to linear independence?
4. Can a transformation be both injective and surjective? If so, what is it called?
5. How does the determinant of a matrix relate to its injectivity or surjectivity?

Tip:

To determine if a matrix transformation is injective, you can check if the matrix has a full column rank. If it does, then the transformation is injective.