LINEAR ALGEBRA. MAT 343

EXAM 2

NAME

ASU ID

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

Set 1:

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

Set 2:

$$\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array}\righ... ...eft( \begin{array}{c} 1 \\ 1 \\ 1 \\ 2 \end{array}\right) .$$

Set 3:

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

 

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

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

3.Calculate rank of the matrix $A.$

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

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

$$\left( \begin{array}{cc} 2 & 9 \\ 4 & 2 \end{array}\right) ,\left( \begin{array}{cc} 0 & 1\\ -4 & 4 \end{array}\right)$$

$$A=\left( \begin{array}{cccc} 3 & 1 & 1 & 1\\ 1 & 3 & 1 & 1... ... 1 & 0 \\ 0 & 0 & 1 \\ -8 & -12 & -6 \\ \end{array}\right)$$

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