Page 344 - 35Linear Algebra

344 Sample Final Exam

3
For any special values of a at which U 6= R , express the subspace U as the
span of the least number of vectors possible. Give the dimension of U for
3
these cases and draw a picture showing U inside R .
7. Vandermonde determinant: Calculate the following determinants

 2 3 
 2  1 x x x
1 x x
2
1 x  1 y y y 3
det , det  1 y y 2  , det  2  .
1 y 2  1 z z z 3
1 z z
1 w w 2 w 3
Be sure to factorize you answers, if possible.
Challenging: Compute the determinant

 2 n−1 
1 x 1 (x 1 ) · · · (x 1 )
1 x 2
 (x 2 ) 2 · · · (x 2 ) n−1
 2 
1 x 3  .
 (x 3 ) · · · (x 3 ) n−1
det 
 . . . .
 . . . . . . . . . 
.
. 
1 x n (x n ) 2 · · · (x n ) n−1
         
 1 3 1 0 0 
3
2
,
,
8. (a) Do the vectors           form a basis for R ?
,
0
0
,
2
1
 
3 1 0 0 1
Be sure to justify your answer.
   
1 4
3
2  
 
4
(b) Find a basis for R that includes the vectors   and   .
3
2
   
4 1
(c) Explain in words how to generalize your computation in part (b) to
n
obtain a basis for R that includes a given pair of (linearly independent)
vectors u and v.
9. Elite NASA engineers determine that if a satellite is placed in orbit starting
at a point O, it will return exactly to that same point after one orbit of the
earth. Unfortunately, if there is a small mistake in the original location of
1
3
the satellite, which the engineers label by a vector X in R with origin at O,
1
This is a spy satellite. The exact location of O, the orientation of the coordinate axes
3
in R and the unit system employed by the engineers are CIA secrets.
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