transformations of curves questions

Please solve these questions. Using attached PDF.

Bonus Problems I: Transformations of curves

Complete the parts (a){(d)

Determine if there exists a linear transformation T : R2 ! R2 that transforms a curve

on the left into a curve on the right. If yes, nd the standard matrix of an appropriate

transformation. If no, explain why.

(a) Circle x2 + y2 = 1 ! ellipse x2 ô€€€ 4xy + 5y2 = 1

For example, the standard matrix of such a T is

No, there is no such transformation, because

Determine if there exists a linear transformation T : R2 ! R2 that transforms a curve

on the left into a curve on the right. If yes, nd the standard matrix of an appropriate

transformation. If no, explain why.

(b) Circle x2 + y2 = 1 ! hyperbola x2 ô€€€ y2 = 1

For example, the standard matrix of such a T is

No, there is no such transformation, because

Determine if there exists a linear transformation T : R2 ! R2 that transforms a curve

on the left into a curve on the right. If yes, nd the standard matrix of an appropriate

transformation. If no, explain why.

(c) Pair of lines x = 1 ! hyperbola x2 ô€€€ y2 = 1

For example, the standard matrix of such a T is

No, there is no such transformation, because

Determine if there exists a linear transformation T : R2 ! R2 that transforms a curve

on the left into a curve on the right. If yes, nd the standard matrix of an appropriate

transformation. If no, explain why.

(d) Hyperbola x2 ô€€€ y2 = 1 ! hyperbola y2 ô€€€ x2 = 1

For example, the standard matrix of such a T is

No, there is no such transformation, because