Plug in r and s into ru + sv and she clude that Span({ủ, v}) = R².

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Answer the question in the picture.

10. Show algebraically that if ủ = (u1, u2) and v = (v1, v2) are vectors in R² that are not
parallel to each other, then any vector (x, y) e R² can be written as a linear combination
of i and v.
Hint: Use the contrapositive that you wrote in Exercise 6 (d) from Section 1.1. Show that
for any x and y, we can solve the vector equation (x, y) = rủ + sv for r and s, in terms of
x, y, u1, u2, V1, and v2. Use the Elimination Method. Show that r and s can be
simplified so that the denominator for both is u¡v2 – u2v1, with no other factors. What
do we know about this denominator? Plug in r and s into ru + sv and show that you get
(x, y) after simplifying.
Next, explain why we can further conclude that Span({ủ, v}) = R².
Transcribed Image Text:10. Show algebraically that if ủ = (u1, u2) and v = (v1, v2) are vectors in R² that are not parallel to each other, then any vector (x, y) e R² can be written as a linear combination of i and v. Hint: Use the contrapositive that you wrote in Exercise 6 (d) from Section 1.1. Show that for any x and y, we can solve the vector equation (x, y) = rủ + sv for r and s, in terms of x, y, u1, u2, V1, and v2. Use the Elimination Method. Show that r and s can be simplified so that the denominator for both is u¡v2 – u2v1, with no other factors. What do we know about this denominator? Plug in r and s into ru + sv and show that you get (x, y) after simplifying. Next, explain why we can further conclude that Span({ủ, v}) = R².
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