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18.024ESG Problem Set 2 Pramod N. Achar
 

Summary: 18.024­ESG Problem Set 2
Pramod N. Achar
Spring 2000
Tuesday
1. (a) Let two lines be given by the following equations:
x = p + ta
y = q + sb
Prove that they intersect if and only if p - q is in the linear span of
a and b.
(b) Suppose we have a k-plane and an l-plane given by the following
equations:
x = p + t1a1 + · · · + tkak
y = q + s1b1 + · · · + slbl
Prove that they intersect if and only if p - q is in the linear span of
{a1, . . . , ak, b1, . . . , bl}.
(c) Determine whether the following 4-plane and 5-plane in R6
intersect.
x = (7, 4, 6, 8, 5, 4) + t1(8, 3, 7, 0, 1, 8) + t2(3, 4, 5, 7, 6, 8)
+ t3(2, 0, 1, 8, 4, 3) + t4(0, 9, 1, 2, 8, 3)
y = (3, 4, 5, 7, 6, 8) + s1(2, 0, 8, 2, 0, 1) + s2(8, 3, 4, 7, 6, 5)

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics