. Let G = GL2(R) and X = R2, and define an action of G on X by ng(x) = gx. As we %3D %3D saw in lectures, this action has two orbits, namely and R? \ {O) Now do the same thing, but replace GL2(R) with the group of upper-triangular invert- ible 2 x 2 matrices. How many orbits are there?

7. Is there a linear transformation T : R³ → R³ such that T If so, what is its matrix?

Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|

(ED). = Let T -x +9y+ 3z - 2x + 4y + 5z 6x 8y + 2z Determine if the given function is a linear transformation. Find a matrix representation A such that T(x) Ax, or indicate that the transformation is not linear. O A = = O Not a linear transformation