Scientific Computing Using Python - PHYS:4905
Practice Exam #1

All work must be shown to receive full credit for each problem.

1. Find the augmented matrix for the system of linear equations below.  Then apply Gaussian elimination and write down your steps as you go.  What is the solution?

(311306322)(xyz)=(122)\begin{pmatrix} 3 & 1 & 1 \\ 3 & 0 & 6 \\ 3 & 2 & 2 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 2 \\ \end{pmatrix}

2. L is a linear operator and  L(22)=(53)L \,\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}    and   L(1-1)=(37)L \, \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} .   Find   L(13)L \, \begin{pmatrix} 1 \\ 1 \end{pmatrix} .


3. Are the following vectors linearly independent and what vector space do they span?

   (111)(221)(110)\begin{pmatrix} 1 \\ 1 \\1 \end{pmatrix} \;\;\; \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \;\;\; \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}


4. What is the determinant of the following matrix?    A=(513742026)A = \begin{pmatrix} 5 & 1 & 3 \\ 7 & 4 & 2 \\ 0 & 2 & 6 \end{pmatrix}

What is the determinant of A2?