Question

In: Advanced Math

Suppose we are given the unit square S in the plane with corners (0, 0), (1,...

Suppose we are given the unit square S in the plane with corners (0, 0), (1, 0), (1, 1) and (0, 1). Let T : R ² → R ² be a linear transformation represented by the matrix A. If T is not onto, will the image of S under the map T be a parallelogram, a line segment, or a point? Be sure to justify your answer.

Expert Solution

We know that the area of the unit square will be

If is not onto then is not trivial meaning (as it must be non-invertible)

This means that whatever the shape of the image of the unit square under is, it is a figure with area

It is impossible it is a parallelogram, but it is possible it is either a line segment or a point (last two are both possible)

For example, transforms the unit square to the unit line segment along the X-axis

And transforms the unit square to the diagonal of the unit square (line segments)

On the other hand, transforms the unit square to the origin (single point)

All the examples above had T as a non-onto transformation

Please do rate this answer positively if you found it helpful. Thanks and have a good day!

Colby Messinger answered 2 years ago

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