The volume scaling factor of the linear transformation described by a matrix has a name. Which?

The volume scaling factor of a linear transformation described by a matrix is known as the determinant of that matrix.

Explanation:

The determinant provides crucial information about the transformation represented by the matrix, including its effect on the volume of geometric shapes. When a matrix transforms a unit volume in space, the determinant indicates how that volume is scaled.

• If the determinant is greater than 1, it means the volume has been expanded.
• If it is less than 1 but greater than 0, the volume has been contracted.
• A determinant of zero indicates that the transformation collapses the volume into a lower dimension, meaning it is not invertible.

Overall, the determinant serves as a key indicator of how linear transformations affect volumes in multi-dimensional spaces.