MAT 343 EXAM 3

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

$$A=\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ ... ... & 0 & 0 & 0 & 1\\ -1 & 0 & 0 & -2 & 0 & 0 \end{array}\right)$$

2. Classify the following second order curves, write the corresponding changes of coordinates and sketch the curves in the coordinate system $(x,y).$

a.

$$x^2 + 4 xy + y^2 + 4x + 4 =0$$

b.

$$5x^2 + 4 xy + 5y^2 + x + y =0$$

3. Write projective canonical forms and the corresponding rotations of the projective plane for the following second order curves.

a.

$$6x^2 + 4 xy + 6y^2 + 4x +4y+ 6 =0$$

b.

$$xy + x + y =0$$