Proving the image of a parallelogram is a parallelogram after a linear transformation.

Question

Let T be an invertible linear transformation from R2 to R2. Let P be a parallelogram in R2 with one vertex at the origin.

Is the image of P a parallelogram? How would I go about finding this out?

I know I need to somehow show more than just the vertices form a parallelogram. Would I need to think about the transformation of the "edge" vectors?

Answer 1

Let $v$ and $w$ two linearly independent vectors. Then the parallelogram $P(v,w)$ with sides $v$ and $w$ is the set of vectors $$ P(v,w)=\{av+bw\mid (a,b)\in[0,1]^2\}. $$ Let $T$ be a linear transformation. Then one has clearly $$ T(P(v,w))=\{aT(v)+bT(w)\mid (a,b)\in[0,1]^2\} $$ which immediately shows that $$ T(P(v,w))=P(T(v),T(w)). $$

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This remains true even if the parallelogram does not have a vertex in the origin. In fact if $P$ is any parallelogram, then $$ P=u+P(v,w) $$ for some vector $u$. But then $$ T(P)=T(u)+P(T(v),T(w)) $$ as above

Answer 2

Hint: linear mappings take lines to lines: $$T({\bf p}+ t {\bf v}) = T{\bf p} + tT{\bf v}.$$ Look at the images of the four sides of your figure by $T$.