Linear algebra 1: cauchy schwarz problem

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Homework Help Overview

The discussion revolves around a linear algebra problem involving the Cauchy-Schwarz inequality and vector norms. The original poster presents a scenario with two vectors, u and v, where the magnitudes and dot product are given, and seeks to find the magnitude of their sum.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants analyze the original poster's attempts to apply the Cauchy-Schwarz inequality and question the validity of their steps. There is a focus on the implications of the dot product and the conditions under which the equality holds.

Discussion Status

Participants are actively engaging with the original poster's reasoning, providing hints and corrections regarding the application of the Cauchy-Schwarz inequality. There is a recognition of the misunderstanding in the application of the inequality and the conditions for equality, leading to further exploration of the problem.

Contextual Notes

Some participants emphasize that the original poster's interpretation of the Cauchy-Schwarz inequality is incorrect, noting that the equality holds only under specific conditions related to the dot product of the vectors.

BWE38
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Homework Statement



If llull = 4, llvll = 5 and u dot v = 10, find llu+vll. u and v are vectors

Homework Equations



llu+vll = llull + llvll cauchy schwarz

The Attempt at a Solution



(1) llu+vll = llull + llvll

(2) (llu+vll)^2 = (llull + llvll)^2

(3) (llu+vll)^2 = llull^2 + 2uv +llvll^2

(4) (llu+vll)^2 = 4^2 + 2(10) +5^2

(5)(llu+vll)^2 = 16 + 20 + 25

(6) sqr(llu+vll)^2 = sqr61

(7) (llu+vll) = 7.81

I think that I am doing it wrong for some reason. Any feed back would be greatly appreciated.
 
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Line (1) is ofcourse not correct.

Hint: square ||u+v||.
 
Last edited:
BWE38 said:

Homework Statement



If llull = 4, llvll = 5 and u dot v = 10, find llu+vll. u and v are vectors

Homework Equations



llu+vll = llull + llvll cauchy schwarz
No, that is NOT correct! The Cauchy-Schwartz in-equality says that
[tex]||u+ v||\le ||u||+ ||v||[/tex]

The Attempt at a Solution



(1) llu+vll = llull + llvll
again, no.

(2) (llu+vll)^2 = (llull + llvll)^2
No.

(3) (llu+vll)^2 = llull^2 + 2uv +llvll^2
That "2uv" (2 time the dot product of u and v) is NOT 0 tells you immediately that ||u+ v|| is NOT equal to ||u||+ ||v||!

(4) (llu+vll)^2 = 4^2 + 2(10) +5^2

(5)(llu+vll)^2 = 16 + 20 + 25

(6) sqr(llu+vll)^2 = sqr61

(7) (llu+vll) = 7.81

I think that I am doing it wrong for some reason. Any feed back would be greatly appreciated.
It looks right to me! What reason do you have to think this is wrong?
 
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HallsofIvy said:
No, that is NOT correct! The Cauchy-Schwartz in-equality says that
[tex]||u+ v||\le ||u||+ ||v||[/tex]


again, no.


No.


That "2uv" (2 time the dot product of u and v) is NOT 0 tells you immediately that ||u+ v|| is NOT equal to ||u||+ ||v||!


It looks right to me! What reason do you have to think this is wrong?

aha! I see what you are saying now. (llu+vll)^2 = uu + uv +vu + vv, which becomes llu+vll^2 = llull^2 + 2uv + llvll^2

llu+vll = llull +llvll, IFF 2uv = 0

Which in this case it isn't. Because uv is equal to 10.
 

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