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Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {Pi + q, P2 + q, P3 + q} is also affinely independent.

Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {Pi + q, P2 + q, P3 + q} is also affinely independent.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {Pi + q, P2 + q, P3 + q} is also affinely independent.
Transcribed Image Text:Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {Pi + q, P2 + q, P3 + q} is also affinely independent.
Expert Solution
Step 1

Here, {p1,p2,p3} is an affinely independent set.

So, {p1,p2,p3} are linearly independent.

Step 2

Consider the set { p1+q, p2+q,p3+q}

Determine the difference of the elements of set.

(p2+q)-(p1+q)= p2-p1

(p3+q)-(p2+q)= p3-p2

For the elements of the set { p1+q, p2+q,p3+q} to be linearly dependent, the differences,

p2-p1 and p3-p2  must be multiple of each other.

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