math54-f22:midterm1-sample
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| ====== Sample Midterm 1 ====== | ====== Sample Midterm 1 ====== | ||
| - | Our midterm will be 50 minutes, hold in discussion session on Monday. 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. | + | 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 ** True or False, | + | ** Problem 1 ** (20 pts) True or False. If you think it is true, give some explanation; |
| - | * Let 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 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 $z^5+5z+3=0$, | ||
| + | * 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 $\det A$. | ||
math54-f22/midterm1-sample.1664610011.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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