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Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors

Instructions

  • This section covers the concepts listed below. 
  • For each concept, there is a conceptual video explaining it followed by videos working through examples and self-assessment questions. 
  • When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page
Systems of Linear Equations
 

There is no conceptual video for this topic.

 

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. Solve the system of equations\[\begin{array}{rrrrrrr}
    x_1 &-& 2x_2 &+& 3x_3 &=&7\\
    -x_1 &+& x_2 &-& 2x_3 &=&-5\\
    2x_1 &-& x_2 &-& x_3 &=&4\\
    \end{array}\]

    The solution is \(x_1=2\), \(x_2=-1\), and \(x_3=1\). Or we can write the solution as
    \[\mathbf{x}=\begin{pmatrix}
    2\\
    -1\\
    1
    \end{pmatrix}
    \]


  2. Discuss the solutions of the system of equations associated with the augmented matrix.\[
    \left( \begin{array}{rrr|r}
    1 & -2 & 3 & b_1\\
    -1 & 1 & -2 & b_2\\
    2 & -1 & 3 & b_3\\
    \end{array}\right)
    \]

    For any value of \(\alpha\), \(x_1=-\alpha-4\), \(x_2=-3+\alpha\), and \(x_3=\alpha\) are a solution of the associated system of equations. Or we can write the solutions as
    \[\mathbf{x}=\alpha
    \begin{pmatrix}
    -1\\
    1\\
    1\\
    \end{pmatrix}
    +
    \begin{pmatrix}
    -4\\
    -3\\
    0
    \end{pmatrix}
    \]
    for any value of \(\alpha\).

 
Linear Independence

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. Are the following vectors linearly independent? If not, find a linear relation between them. 
    \[
    \mathbf{x_1}=\begin{pmatrix} 1\\ 2\\ -1 \end{pmatrix}, \quad
    \mathbf{x_2}=\begin{pmatrix} 2\\ 1\\ 3 \end{pmatrix},
    \quad
    \mathbf{x_3}=\begin{pmatrix} -4\\ 1\\ -11 \end{pmatrix}
    \]

    The vectors are not linearly indepent since \(-2\mathbf{x_1}+3\mathbf{x_2}+\mathbf{x_3}=\mathbf{0}\). Or we can write that \(\mathbf{x_3}=2\mathbf{x_1}-3\mathbf{x_2}\). 

 
Eigenvalues and Eigenvectors

Exercises


Directions: There are no exercises for this topic.
 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. How do you write the general solution of a linear system of ODEs when there is a distinct pair of real eigenvalues?