MAT 342 EXAM 3 (REVIEW)

NAME:______________________

ASU ID:_____________________

1. Find the canonical form and the corresponding change of coordinates for the following linear operators.

$$A=\left( \begin{array}{cccc} 4 & 1 & 1 & -6\\ 1 & 4 & 1... ...0 & 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$$

2. Classify the following second order curve, find the corresponding affine change of coordinates and sketch it in the coordinate system $(x,y).$

$$\mbox{(a)}\;\;\;x^2 +2\sqrt{2}xy +2 y^2 + x + y = 0$$

$$\mbox{(b)} \;\;\; 2x^2 +4xy - y^2 + 2x + y +1 = 0$$

3. Calculate the projective canonical form for the second order curve and the corresponding motion of the projective plane.

$$\mbox{(a)}\;\;\; x^2 + 6 xy + y^2 + 6x + 6y+ 1 =0$$

$$\mbox{(b)} x^2 + 4 xy + 4 y^2 + 4x + 4 =0$$