Invariants Under Transformation Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Under a shear transformation defined by

(x, y) \to (x + 2y, y)

, determine whether the area of a unit square with vertices

(0,0)

(1,0)

(1,1)

(0,1)

is preserved.

1
Apply the shear to each vertex: $(0,0) \to (0,0)$ , $(1,0) \to (1,0)$ , $(1,1) \to (3,1)$ , $(0,1) \to (2,1)$ . The image is a parallelogram.
2
The area of the parallelogram can be found using the cross product of two edge vectors: $\vec{u} = (1,0)$ and $\vec{v} = (2,1)$ . Area $= |1 \cdot 1 - 0 \cdot 2| = 1$ . The area equals the original, so area IS invariant under this shear.

Answer

\text{Yes, the area is invariant. The sheared parallelogram has area } 1.

Shear transformations have a transformation matrix with determinant

1

(since

\det\begin{pmatrix}1&2\\0&1\end{pmatrix}=1

). Any linear transformation with

|\det|=1

preserves area. Shears change shape but not area — they are area-preserving but not distance-preserving.

About Invariants Under Transformation

A property of a function is invariant under a transformation if it remains unchanged after the transformation is applied to the function.

A triangle with vertices at [formula], [formula], and [formula] is translated by the vector [formula

A rectangle with vertices [formula], [formula], [formula], [formula] is dilated by a scale factor of

A figure is reflected over the [formula]-axis. Determine whether each property is invariant: (a) sid