Linear Algebra
Test #1
March 21, 2003
Name____________________
R.  Hammack
Score ______


(1) (9 points) This problem concerns the vectors [Graphics:Images/T1S03Asol_gr_1.gif],  [Graphics:Images/T1S03Asol_gr_2.gif],  [Graphics:Images/T1S03Asol_gr_3.gif].
(a) [Graphics:Images/T1S03Asol_gr_4.gif]

(b) [Graphics:Images/T1S03Asol_gr_6.gif]

(c) Two of the vectors u, v, and w are orthogonal. Which two, and why?


(2) (10 points) Find the distance between the point [Graphics:Images/T1S03Asol_gr_7.gif]in  [Graphics:Images/T1S03Asol_gr_8.gif] and the plane [Graphics:Images/T1S03Asol_gr_9.gif].




(3) (10 points) Find the solution of the following linear system. Write the solutions in vector form.

[Graphics:Images/T1S03Asol_gr_16.gif]
[Graphics:Images/T1S03Asol_gr_17.gif]

 



(4)
(a) (10 points) Consider the set of vectors  [Graphics:Images/T1S03Asol_gr_26.gif]  in [Graphics:Images/T1S03Asol_gr_27.gif].  Is this a line or a plane?  If it is a line, write its equation in vector from. If it is a plane, write its equation in normal form.

 




(b) (10 points) Consider the set of vectors  [Graphics:Images/T1S03Asol_gr_31.gif]  in [Graphics:Images/T1S03Asol_gr_32.gif].  Is this a line or a plane?  If it is a line, write its equation in vector from. If it is a plane, write its equation in normal form.

 

 




(5) (10 points) Suppose A, B and X are invertible n-by-n matrices. Solve the following equation for X.

[Graphics:Images/T1S03Asol_gr_36.gif]

 


(6) This problem concerns the vectors [Graphics:Images/T1S03Asol_gr_47.gif], [Graphics:Images/T1S03Asol_gr_48.gif], [Graphics:Images/T1S03Asol_gr_49.gif].

(a) (10 points) Are these vectors linearly independent or linearly dependent? Show your work.




(b) (5 points) Is the span of these vectors equal to [Graphics:Images/T1S03Asol_gr_52.gif]?  Why or why not?




(7) (10 points) Suppose A is a 3-by-4 matrix. Are the columns of A linearly independent, linearly dependent, or is there not enough information to say?  Explain.

 

 


(8) (10 points) This problem concerns the invertible matrix [Graphics:Images/T1S03Asol_gr_55.gif]
(a) Find [Graphics:Images/T1S03Asol_gr_56.gif].



(b) (6 points)  Use your answer to part (a) above to find a solution to the equation [Graphics:Images/T1S03Asol_gr_59.gif].