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:

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

Set 2:

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

Set 3:

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

 

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

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

3.Calculate the 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.

$$\left( \begin{array}{cc} 3 & 2 \\ -4 & -3 \end{array}\right) ,\left( \begin{array}{cc} 0 & 1\\ -9 & 6 \end{array}\right),$$

$$\left( \begin{array}{ccc} 1 & 4 & 4\\ 4 & 1 & 4\\ 4 & 4 & ... ...1 & 0\\ 0 & 0 & 0 & 1\\ -4 & 0 & 5 & 0 \end{array}\right),$$

5. Solve the following three diagonal system with backtracking approach (For instructions see the method "solveThreeDiagonal" in LinearSystem.java).

$$\left( \begin{array}{ccccc} 2 & 1 & 0 & 0 & 0 \\ 1 & -2 & 1... ...eft( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1 \end{array}\right)$$