Page 840 - Advanced_Engineering_Mathematics o'neil

820 Answers to Selected Problems

so A has rank 3. (b) < 0,4,3 >, < 6,1,0 > and < 2,2,2 > form a basis for the row space of A. The column space has
basis consisting of
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 4 3
⎝ 6 ⎠ , ⎝ 1 ⎠ , and ⎝ 0 ⎠ .
2 2 2
11.
⎛ ⎞
1 0 0
A R = ⎝ 0 1 0 ⎠ ,
0 0 1
so rank(A) = 3. (b) The row space has basis consisting of < −3,2,2 >, < 1,0,5 > and < 0,0,2 >. The column space
has basis consisting of
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−3 2 2
⎝ 1 ⎠ , ⎝ 0 ⎠ , and ⎝ 5 ⎠ .
0 0 2
13.
⎛ ⎞
1 0 −11
A R = ⎝ 0 1 −3 ⎠ ,
0 0 0
so a has rank 2. (b) < −2,5,7 > and < 0,1,−3 > form a basis for the row space. (c) The column space has basis
consisting of
⎛ ⎞ ⎛ ⎞
−2 5
⎝ 0 ⎠ and ⎝ 1 ⎠ .
4 11
15.
rank(A) = dim row space of A
= dim col space of A = dim row space of A T
t
= rank(A ).

Section 7.5 Homogeneous Systems of Equations
1.
−1 1
⎛ ⎞ ⎛ ⎞
⎜ 1 ⎟ ⎜−1⎟
1 0
α ⎝ ⎠ + β ⎝ ⎠
0 1
The solution space has dimension 2.
3.
⎛ ⎞
0
α ⎝ 0 ⎠
0
(only the trivial solution). The solution space has dimension 0.
5.
−9/4
⎛ ⎞
⎜−7/4⎟
⎜ ⎟
α ⎜−5/8⎟
13/8
⎝ ⎠
1
The solution space has dimension 1.

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October 14, 2010 17:50 THM/NEIL Page-820 27410_25_Ans_p801-866