CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-256 Review Exam 2, Spring 2004

1. Determine if a vector $\vec{w}$ is in the span of vectors { $\vec{v_1}, \ldots, \vec{v_r}\}$.
   Example: Let $\vec{w}=[5,2,-2]$, $\vec{v_1}=[1,4,2]$, and $\vec{v_2}=[3,0,7].$ Determine if $\vec{w}$ is in the span of { $\vec{v_1},\vec{v_2}\}$.

2. Find the determinant of an $n\times n$ matrix.
   Example:
   
   $$\det \left[ \begin{array}{rrrr} 2 & 0 & -1 & 1 \\ 0 & 3 & 1 & 0 \\ 1 & -1 & 0 & 0 \\ 2 & 0 & 0 & 1 \end{array}\right].$$
   
   

3. Determine if $\{\vec{v_1}, \ldots, \vec{v_r}\}$ is linearly independent.
   Example: Let $\vec{v_1}=[1,4,3,1]$, $\vec{v_2}=[2,-3,0,1]$, and $\vec{v_3}=[1,-18,-9,-1].$ Determine if $\{\vec{v_1}, \vec{v_2}, \vec{v_3}\}$ is linearly independent.

4. Solve a system of equations.
   Example: Find all solutions to the following system of equations by using row operations on an appropriate augmented matrix. Please indicate the specific row operation you use in each step.
   
   $$\begin{array}{rrrrrrrrrrr} x_1 & + & 3x_2 & & & + & 2x_4 & -... ...= & -2 \\ & & & & x_3 & + & 4x_4 & + & 2x_5 = & 1 \end{array}$$
   
   

5. Determine if a matrix $A$ is invertible. If it is, find $A^{-1}$, and use it to solve an equation of the form $A\vec{x}=\vec{b}$. Example: Let
   $$\begin{eqnarray*} A & = & \left[ \begin{array}{rrr} 1 & 3 & 0 \\ 1 & -1 & 1 \\ 2 & 0 & 1 \end{array}\right]. \end{eqnarray*}$$
   
   
   
   
   
   Find $A^{-1}$ (if it exists) and use it to solve $A\vec{x}=\vec{b}$, where $\vec{b}=[3,1,-2]^T$

6. Take the dot product of vectors, and use it to determine if vectors are othogonal.
   Example: For what values $t$ are the vectors $\vec{v}=\langle t,2,-t \rangle$ and $\vec{w}=\langle t,1,3 \rangle$ orthogonal?

7. Find the cross product of two vectors.
   Example: Let $\vec{v}=\langle 2,1,3 \rangle$, $\vec{w}=\langle -2,1,4 \rangle$. Compute $\vec{v} \times \vec{w}$.

8. Find the equation of a line through two points, or a line through a point in a specified direction $\vec{v}$.
   Example: Find the equation of the line through $P_0(4,2,-3)$ and $P_1(-2,2,5)$.

9. Find the equation of the plane containing a point $P$ and with given normal vector $\vec{n}$.
   Example: Find the equation of the plane containing the point $P(2,-1,3)$ with normal vector $\vec{n}=\langle 5,-4 2 \rangle$.