Sample Midterm 1

Our midterm will be 50 minutes, hold in discussion session on next Wednesday (Oct 5). The test range is Givental [LA] Ch 1 and Ch 2, and Ch 3 with the concept of basis, linear independence and span. We will have 5 problems, each worth 20 points.

Problem 1 (20 pts) True or False. If you think it is true, give some explanation; if you think it is false, give a counter-example.

• Let the conic curve $C$ be given by the equation $a x^2 + b xy + cy^2=1$, where $a,b,c$ are real numbers. If $a,b,c$ are all positive, then $C$ must be an ellipse.
• Let $T: \R^2 \to \R^2$ be a linear transformation. If $T( (1,0)) \neq (0,0)$ and $T( (0,1) ) \neq (0,0)$, then $T$ must be invertible.
• Let $z_1, \cdots, z_5$ be the roots of polynomial equation $z^5+5z+3=0$, then $z_1+\cdots+z_5=0$.
• Let $A$ be a 3 by 3 matrix. If $A^2=0$, then $A$ is the zero matrix.

Problem 2 (20 pts) $$ \begin{pmatrix} 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr \end{pmatrix}^2=? $$

$$ \det \begin{pmatrix} 1 & 1 & 1 \cr 0 & 2 & 1 \cr 0 & 3 & 4 \cr \end{pmatrix}=? $$

Problem 3 (20 pts) Represent the bilinear form $B( (x_1,x_2), (y_1,y_2) ) = 2 x_1(y_1+y_2)$ in $\R^2$ as the sum $S+A$ of a symmetric and an anti-symmetric ones.

Problem 4 (20 pts) What is the length of the permutation $\begin{pmatrix} 1 & 2 & 3 & 4 \cr 2 & 4 & 3 & 1 \end{pmatrix}$?

Problem 5 (20 pts) Let $A$ be a size $n$ matrix. Express $\det(adj(A))$ using $\det A$.