linear-guest

344 Sample Final Exam
For any special values of a at which U ≠ R 3 , express the subspace U as the
span of the least number of vectors possible. Give the dimension of U for
these cases and draw a picture showing U inside R 3 .
7. Vandermonde determinant: Calculate the following determinants
det
( ) 1 x
1 y
⎛
1 x x 2 ⎞
, det ⎝1 y y 2 ⎠ ,
1 z z 2
Be sure to factorize you answers, if possible.
Challenging: Compute the determinant
⎛
1 x x 2 x 3 ⎞
det ⎜1 y y 2 y 3
⎟
⎝1 z z 2 z 3 ⎠ .
1 w w 2 w 3
⎛
1 x 1 (x 1 ) 2 · · · (x 1 ) n−1 ⎞
1 x 2 (x 2 ) 2 · · · (x 2 ) n−1
det
1 x 3 (x 3 ) 2 · · · (x 3 ) n−1
.
⎜
⎝
.
. . . ..
⎟
. ⎠
1 x n (x n ) 2 · · · (x n ) n−1
⎧⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫
⎨ 1 3 1 0 0 ⎬
8. (a) Do the vectors ⎝2⎠ , ⎝2⎠ , ⎝0⎠ , ⎝1⎠ , ⎝0⎠
⎩
⎭ form a basis for R3 ?
3 1 0 0 1
Be sure to justify your answer.
⎛ ⎞ ⎛ ⎞
1 4
(b) Find a basis for R 4 that includes the vectors ⎜2
⎟
⎝3⎠ and ⎜3
⎟
⎝2⎠ .
4 1
(c) Explain in words how to generalize your computation in part (b) to
obtain a basis for R n 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
the satellite, which the engineers label by a vector X in R 3 with origin 1 at O,
1 This is a spy satellite. The exact location of O, the orientation of the coordinate axes
in R 3 and the unit system employed by the engineers are CIA secrets.
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