Question

Let \( M_{n \times p}(\mathbb{R}) \) be the subspace of \( M_{n \times p}(\mathbb{R}) \) defined by \[ V = \{ X \in M_{n \times p}(\mathbb{R}) : TX = 0 \}. \] Then the dimension of \( V \) is

• A. \( pn - \text{rank}(T) \)
• B. \( mn - p \text{rank}(T) \)
• C. \( p(m - \text{rank}(T)) \)
• D. \( p(\text{n} - \text{rank}(T)) \)

Hint:

In linear algebra, the rank-nullity theorem is crucial for understanding the dimensions of the kernel and image of linear transformations.

Answer 1

The Correct Option is D

Step 1: Understanding the problem.
We are given a subspace \( V \) of matrices in \( M_{n \times p}(\mathbb{R}) \), where the matrices satisfy the condition \( TX = 0 \), meaning that the image of the matrix \( X \) under the linear transformation \( T \) is zero. This implies that \( X \) is in the null space of \( T \).

Step 2: Analyzing the options.
The rank-nullity theorem states that the dimension of the null space of a linear transformation is given by the total number of columns minus the rank of the transformation. Here, the number of columns in \( X \) is \( p \), and we are interested in the dimension of the space of all possible matrices that satisfy \( TX = 0 \). Thus, the dimension of the subspace \( V \) is \( pn - \text{rank}(T) \).

Step 3: Conclusion.
The correct answer is (A) \( pn - \text{rank}(T) \), as this follows directly from the rank-nullity theorem.