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(MATH111)[2010](s)final~cs_zxxab^_10439.pdf
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FINAL EXAM MATH111 Linear Algebra Spring 2010
There are 5 problems. Show the working steps of your answers for full credits.
1. (25 points) Let vectors , , and . ........... 413 u
............ 012 v1
............ 221 v2
(1) Find the inner products and . 1vu.
2vu.
(2) Find and . 1v
2v
(3) Show that and are orthogonal to each other. 1v
2v
(4) Let . Find and the distance from to . ..21vvSpan,.W
uprojW
u
W
Solution:
(1) ..516041123vu1.............
. ..3823242113vu2.............
(2) ..5012v2221.....
..3221v2222.....
(3) ....0202112vv21..........
Thus and are orthogonal to each other. 1v
2v
(4)
................................................................... 323535 221 31 012 221 93 012 55vvvuvvvuuproj22221211W
. Thus the distance from to ................................... 3103834 323535 413 uprojuW
u
523103834uproju222........................WW
2. (25 points) Consider the matrix
. ................. 0111101111011110 A
The characteristic equation of is . A
....0313.....
(1) Find the eigenvalues of the matrix .
(2) For each eigenvalue, find a basis for the corresponding eigenspace.
(3) Orthogonally diagonalize the matrix .
Solution:
(1) Characteristic equation
Eigenvalues: (multiplicity 3), . 11..
32...
(2) For 1..
..................................... 0000000000001111 1111111111111111 IA.
..0x..IA.
Solution ................................................................... 1001 0101 0011 4324324324321xxxxxxxxxxxxx
A basis for the corresponding eigenspace is . ................................................... 1001 0101 0011 ,,
For 3...
................................ 0000110010101001 3111131111311113 IA.
Solution .......................................... 1111 444444321xxxxxxxxx
A basis for the corresponding eigenspace is . ............................ 1111
(3) A basis for the eigenspace of is , where 11..
..321x,xx,
, , ............. 0011 x1
............. 0101 x2
.............. 1001 x3
A basis for the eigenspace of is , where 32...
..4x
. ............... 1111 x4
Perform Gram-Schmidt process to obtain an orthogonal basis for the eigenspace of .
.............. 0011 xv11
. Choose ........................................... 012121 0011 21 0101 vvvxxv1211222
................ 0211 2vv22
. ................................................................. 1313131 0211 61 0011 21 1001 vvvxvvvxxv222231211333
Normalizing these vectors
............................. 002121 0011 21vv1u111
................................. 0626161 0211 61vv1u222
............................... 23321321321 1313131 12311vv1u333
. ............................... 21212121 1111 41xx1u444
Let ...................... 21230021321620213216121213216121 uuuu 4321P
.............. 30000