Linear Algebra Exercises-n-Answers.pdf

176 Linear Algebra, by Hefferon
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2 The number of operations depends on exactly how the operations are carried out.
(a) The determinant is −11. To row reduce takes a single pivot with two multiplications (−5/2
times 2 plus 5, and −5/2 times 1 plus −3) and the product down the diagonal takes one more
multiplication. The permutation expansion takes two multiplications (2 times −3 and 5 times 1).
(b) The determinant is −39. Counting the operations is routine.
(c) The determinant is 4.
3 One way to get started is to compare these under Octave: det(rand(10));, versus det(hilb(10));,
versus det(eye(10));, versus det(zeroes(10));. You can time them as in tic(); det(rand(10));
toc().
4 This is a simple one.
DO 5 ROW=1, N
PIVINV=1.0/A(ROW,ROW)
DO 10 I=ROW+1, N
DO 20 J=I, N
A(I,J)=A(I,J)-PIVINV*A(ROW,J)
20 CONTINUE
10 CONTINUE
5 CONTINUE
5 Yes, because the J is in the innermost loop.
Topic: Projective Geometry
1 From the dot product
⎛
⎞
0 = ⎝ 1 0⎠ L 1 L 2
)
L 3 = L1
0
we get that the equation is L 1 = 0.
2 (a) This determinant
1 4 x
0 =
2 5 y
= −3x + 6y − 3z
∣3 6 z∣ shows that the line is L = ( ⎛ ⎞
−3 6 −3 ) .
(b) ⎝ −3
6 ⎠
−3
3 The line incident on
⎛ ⎞ ⎛ ⎞
u 1
v 1
u = ⎝u 2
⎠ v = ⎝v 2
⎠
u 3 v 3
can be found from this determinant equation.
u 1 v 1 x
0 =
u 2 v 2 y
∣u 3 v 3 z∣ 2v 3 − u 3 v 2 ) · x + (u 3 v 1 − u 1 v 3 ) · y + (u 1 v 2 − u 2 v 1 ) · z
The equation for the point incident on two lines is the same.