Given a vector with cartesian components (x, y), if scaling is done with matrix \(\left[\begin{array}{cc} 0.5 & 0 \\ 0 & 1.5 \end{array}\right]\), which of the following are true.

A. Decreases the vertical by three halves

B. Increases the vertical by three halves

C. Doubles the horizontal

D. Halves the horizontal

Choose the correct answer from the options given below:

The correct answer is B and D only

Key PointsIn linear algebra, scaling is a linear transformation that enlarges or diminishes objects. The scaling matrix is diagonal with scaling factors on the diagonal. It appears you're giving entries of the diagonal matrix.

Let's denote the vector as V = [x, y], and the scaling matrix as M = [0.5, 1.5]. When you multiply the vector V by the matrix M, you get a new vector V' as follows:

\(V' = M \cdot V = \begin{bmatrix} 0.5 & 0 \\ 0 & 1.5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\)

The resulting vector is:

\(V' = \begin{bmatrix} 0.5 \cdot x \\ 1.5 \cdot y \end{bmatrix} \)

This represents a transformation that scales the width (x component) by a factor of 0.5 (which means halving) and the height (y component) by a factor of 1.5 (which means increasing by 1.5x or three halves).

Therefore, the correct answers are: Halves the horizontal and Increases the vertical by three halves.