Math 240,

the course of Dr. Mihailovs


Midterm 1

July 9, 1998

  1. Let ${\bf a}=\begin{pmatrix} 1\\ -2\\ 1 \end{pmatrix}, \thickspace {\bf b}=\begin{pmatrix} 1\\ 1\\ -1 \end{pmatrix}$. Find ${\bf a\cdot b}$ and ${\bf a\times b}$.

  2. Let $A=\begin{pmatrix} 1 & 2\\ -1 & 1 \\ 1 & 2 \end{pmatrix}, \thickspace B=\begin{pmatrix} 1& -1\\ 1 & 1 \end{pmatrix}$. Find AB and B-1.

  3. Find the equation of the plane through $\begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}$ perpendicular to the line $x=3+2t,\thickspace y=1-t,\thickspace z=t$.

  4. Let ${\bf a}=\begin{pmatrix} 3\\ -6\\ 9 \end{pmatrix}$ and ${\bf b}=\begin{pmatrix} 2\\ 1\\ 2 \end{pmatrix}$. Find a unit vector parallel to ${\bf b}$ and the projection of ${\bf a}$ in the direction of ${\bf b}$.

  5. Let $A=\begin{pmatrix} 1 & 1 & 1\\ 1 & 2 & 3\\ 1 & 4 & 6 \end{pmatrix}$. Write out the Laplace expansion of $\det A$ by column 2 (do not compute the determinant).

  6. Compute $ ({\bf e}_1+2{\bf e}_2)\wedge (3{\bf e}_1+4{\bf e}_2)$.

  7. Let TA be a linear mapping $T_A: x\mapsto Ax$ where $A=\begin{pmatrix} 1 & 2\\ 2 & 1 \end{pmatrix}$. Sketch the image under TA of the unit square ${\bf D}$ spanned by ${\bf e}_1,\thickspace {\bf e}_2$ and oriented counterclockwise. Does TA preserve or reverse the orientation?

  8. Find a parametric equation for the line through $\begin{pmatrix} 1\\ 1\\ -1 \end{pmatrix}$ and $\begin{pmatrix} -1\\ 2\\ 3 \end{pmatrix}$.

  9. Is the line x+y=-1 a subspace of $ \mathbb {R} ^2$? Is the half-space $z\geq 0$ a subspace of $ \mathbb {R} ^3$?

  10. Find the eigenvalues and corresponding eigenvectors for the symmetric matrix $A=\begin{pmatrix} 0 & -1\\ -1 & 0 \end{pmatrix}$.

  11. With A as above, find an orthogonal matrix B such that B-1AB is diagonal.

  12. Compute the area of the parallelogram spanned by ${\bf a}=\begin{pmatrix} 1\\ -2\\ 1 \end{pmatrix}$ and ${\bf b}=\begin{pmatrix} 1\\ 1\\ -1 \end{pmatrix}$ (same as in Problem 1).

  13. In each case below determine whether the given vectors are linearly dependent or linearly independent (don't compute):

  14. Suppose that one can reduce the augmented coefficient matrix of the system $A{\bf x}={\bf 0}$ to the eschelon form $\begin{pmatrix} 1 & 2 & 3 & 4 & 0\\ 0 & 0 & 1 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$. What is the dimension of the space of solutions of the given system?

  15. With regard to the usual basis ${\bf e}_1=\binom{1}{0},\thickspace {\bf e}_2=\binom{0}{1}$ in $ \mathbb {R} ^2$, P has coordinates $\binom{2}{1}$. What are the coordinates of P with regard to a new basis ${\bf b}_1= \binom{1}{1},\thickspace {\bf b}_2=\binom{-1}{1}$ ? (Hint: use Problem 2.)

  16. Suppose that A is a $2\times 2$ matrix and that the following sequence of basic operations reduces A to I:

    \begin{displaymath} A\xrightarrow[\frac{1}{2} \row 1]{\row 1}A_1\xrightarrow[\row 2-\row 1] {\row 2}I\end{displaymath}

    Find the elementary matrices which produce these operations. Find A-1.

  17. Let $T_A: \mathbb {R} ^4\rightarrow \mathbb {R} ^3,\quad {\bf x}\mapsto A{\bf x}$ where $A=\begin{pmatrix} 1 & 2 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}$. Find a basis for the range of TA and give its dimension. Find the dimension of the null space of TA.

  18. Let $A=\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}$. Write down eAt (just give the answer.)

  19. With C=eAt as above, what does the map ${\bf x}\mapsto C{\bf x}$ do to $ \mathbb {R} ^2$?

Copyright © Aleksandrs Mihailovs 7/9/1998