TEST 1 (LINEAR ALGEBRA)

ASU ID:..................................NAME:..................................

1. Find all of the solutions, if any exist, for the following systems.

System 1:

$$\begin{eqnarray*} &&2\cdot x_1+3x_2=1 \\ &&x_1+3\cdot x_2+x_3=1 \\ &&x_2+4\... ...1 \\ &&x_3+3\cdot x_4+x_5=1 \\ &&3x_4+2\cdot x_5=1. \\ && \end{eqnarray*}$$

System 2:

$$\begin{eqnarray*} 4\cdot x_1+5\cdot x_2+x_3 &=&0 \\ 3\cdot x_1+4\cdot x_2+x_3 &=&0 \\ 9\cdot x_1+4\cdot x_2+x_3 &=&6. \end{eqnarray*}$$

System 3:

$$\begin{eqnarray*} x_1+ x_2+x_3 +x_4 &=&2 \\ x_1- x_2+4x_3-3x_4 &=&0 \\ x_1+ x_2+16x_3+9x_4 &=&2 \\ x_1- x_2+64x_3-27x_4 &=&0 \end{eqnarray*}$$

2.Given matrices $A,\;\;B$ calculate $A\cdot B.$

a) $A=\left( \begin{array}{cc} 3 & 4 \\ 1 & 0 \\ -1 & 5 \end{array} \right... ...\begin{array}{cccc} 1 & 3 & -5 & 1 \\ 2 & -4 & 2 & 0 \end{array} \right) .$

b) $A=\left( \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 1... ...& 2 \end{array} \right) . \begin{array}{cc} & \\ & \\ & \end{array}$

3.Given a matrix $A$ find $A^{-1}.$

a) $A=\left( \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \end{array} \right) .\;\;$b) $\;A=\left( \begin{array}{ccccc} 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 \\ ... ...1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 \end{array} \right) .$

4.Use Cramer's Rule to find the solution of the system

$$\begin{array}{cccccccc} 2x_1+ & x_2+ & x_3+ &x_4 + &x_5 + ... ... x_1+ & x_2+ & x_3+ &x_4 + &x_5 + &2x_6 & = & 7 \end{array}$$

5.Calculate a basis for $Im(A)$ and a basis for $ker(A).$

$$A=\left( \begin{array}{cccccc} 2 & 1 & 3 & 0 & 1 &1\\ ... ...& 2 & 1 & -1\\ 1 & 0 & 1 & 0 & 2 & 1 \end{array} \right)$$

6. Find all solutions (if any exist) for the following linear system.

$$\begin{eqnarray*} x_1+ x_2+x_3 +x_4 &=&0 \\ x_1- x_2+3x_3 &=&0 \\ x_1+ x_2-2x_3+x_4 &=&0 \\ x_1- x_2+7x_3 +x_5 &=&0 \end{eqnarray*}$$

7. Classify each of the following sets of vector fields as linearly dependent or independent.

Set 1:

$$\left( \begin{array}{c} 0 \\ 0 \\ 1 \\ 1 \\ 1 \e... ...ay}{c} 1 \\ 0 \\ 1 \\ 1 \\ 1 \end{array} \right) .$$

Set 2:

$$\left( \begin{array}{c} 3 \\ 1 \\ 1 \\ 3 \end{arra... ...gin{array}{c} 1 \\ 1 \\ 1 \\ 2 \end{array} \right) .$$

Set 3:

$$\left( \begin{array}{c} 0 \\ 0 \\ 3 \\ 1 \end{arra... ...gin{array}{c} 0 \\ 1 \\ 4 \\ 2 \end{array} \right) .$$