What is the transformation matrix for a dilation with a scale factor of k?

The transformation matrix for a dilation with a scale factor of ( k ) uniformly scales both the x and y coordinates of a point in space. When applying a dilation, each point ((x, y)) is transformed to a new point ((kx, ky)). This scaling effect is uniformly applied in all directions determined by the scale factor ( k ).

The correct transformation matrix captures this effect in the following form:

[

\begin{bmatrix}

k & 0 \

0 & k

\end{bmatrix}

]

In this matrix, the diagonal elements both represent the scale factor ( k ), which means that the x-coordinate is multiplied by ( k ) and the y-coordinate is also multiplied by ( k ). The off-diagonal elements being zero indicates that there is no shearing or other types of transformation occurring, only pure dilation.

The other choices provided do not fulfill the criteria of a dilation with uniform scaling. For example, one might only scale one dimension while leaving the other unchanged or misrepresent the transformation entirely. Hence, the correct matrix effectively indicates a consistent expansion or contraction of the figure in all directions, aligning perfectly with the definition and characteristics of a dilation in a two-dimensional space.