LINEAR ALGEBRA. MAT 343

EXAM 2 (due Tue.11/02/10 (computer lab))

NAME

ASU ID

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

Set 1:

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

Set 2:

$$\left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\righ... ...eft( \begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \end{array}\right) .$$

Set 3:

$$\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array}\righ... ...t( \begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \end{array}\right) .$$

 

2.Given a linear operator $A$ find a basis of its kernel $ker(A)$ and a basis for its image $im(A).$

$$A= \left( \begin{array}{cccccc} 1 & 2 & 1 & 2 & 3 & 9\\ -1 ... ...1 & -1 & -1 & -4 \\ 1 & 3 & 2 & 3 & 4 & 13 \end{array}\right)$$

3.Calculate the rank of the matrix $A.$

$$A= \begin{array}{ccc} \left( \begin{array}{c} 2 \\ 1 \\ 1 \... ...\ 5 \\ 5 \end{array}\right) . & & \\ & & \\ & & \end{array}$$

4. Find the canonical forms for the following linear operators and the matrices for the corresponding changes of coordinates.

$$\left( \begin{array}{cc} 6 & 2 \\ -2 & 2 \end{array}\right) ,\left( \begin{array}{cc} 0 & 1\\ -17 & 8 \end{array}\right)$$

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

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

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