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:

$$\left( \begin{array}{c} 4 \\ 4 \\ 8 \\ 1 \\ 1 \... ...ay}{c} 9 \\ 0 \\ 1 \\ 1 \\ 1 \end{array} \right) .$$

Set 2:

$$\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 6 \end{arr... ...gin{array}{c} 1 \\ 1 \\ 1 \\ 0 \end{array} \right) .$$

Set 3:

$$\left( \begin{array}{c} 1 \\ 0 \\ 9 \\ 1 \end{arr... ...in{array}{c} 1 \\ 0 \\ 8 \\ 12 \end{array} \right) .$$

 

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

$$A= \left( \begin{array}{ccccc} 2 & -1 & 1 & 2 & 1 \\ 2 & 1... ... 2 & -1 & 1 & 2 & 1 \\ 2 & 1 & 1 & 4 & 5 \end{array}\right)$$

3.Calculate rank of the matrix $A.$

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

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

$$\left( \begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right) ,\left( \begin{array}{cc} 0 & 1\\ -5 & 2 \end{array}\right)$$

5. Solve the following three diagonal system with backtracking approach.

$$\left( \begin{array}{ccccc} 1 & 1 & 0 & 0 & 0 \\ 1 & -3 & 1... ...eft( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1 \end{array}\right)$$