Image of a linear transformation

[morsel]

Homework Statement

Let T be an invertible linear transformation from R2 to R2. Let P be a parallelogram in R2. Is the image of P a parallelogram as well? Explain.

P is a parallelogram in the first quadrant without any specified points.

Homework Equations

The Attempt at a Solution

I'm not sure how to begin when the points are not specified. Any hints?

 

[HallsofIvy]

Linear transformation map straight lines into straight lines. Since this linear transformation is invertible, it does NOT map different points to the same point so it maps the four different lines forming the parallelogram into four different lines: it maps quadrilaterals into quadrilaterals. Further, because a linear transformation acts on vectors, not points, it maps parallel vectors into parallel vectors.

 

[morsel]

Thank you so much for the explanation, HallsofIvy!