(a) Use the general properties of an inner product (·, ·) on a vector space V, with induced norm ||u|| = V(u, u), to show the Cauchy-Schwarz inequality: For any u, v € V, |(u, v)| < || u|| ||v||- (CS) [Hint: For eacht e R, we have (u+tv, u+tv) > 0 (justify this). Expand this inner prod- uct to get an expression that is quadratic in t; show that the condition for this quadratic to remain no-negative for all t is equivalent to (CS). Of course for the geometric dot product, involving the cosine of an angle, (CS) can be shown very easily...] (b) For any norm || · || induced by an inner product, as in (a) above, prove the triangle inequality: For any u, v e V, ||u + v|| < ||u|| + ||v||, (A) with equality iff u and v are orthogonal. [Hint: Square both sides. Of course for || - || to be a norm it must satisfy (A); this result shows that for any inner product, the quantity /(u, u) indeed defines a norm.]

(a) Use the general properties of an inner product (·, ·) on a vector space V, with induced norm ||u|| = V(u, u), to show the Cauchy-Schwarz inequality: For any u, v € V, |(u, v)| < || u|| ||v||- (CS) [Hint: For eacht e R, we have (u+tv, u+tv) > 0 (justify this). Expand this inner prod- uct to get an expression that is quadratic in t; show that the condition for this quadratic to remain no-negative for all t is equivalent to (CS). Of course for the geometric dot product, involving the cosine of an angle, (CS) can be shown very easily...] (b) For any norm || · || induced by an inner product, as in (a) above, prove the triangle inequality: For any u, v e V, ||u + v|| < ||u|| + ||v||, (A) with equality iff u and v are orthogonal. [Hint: Square both sides. Of course for || - || to be a norm it must satisfy (A); this result shows that for any inner product, the quantity /(u, u) indeed defines a norm.]

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

1. Norms and inner products – Cauchy-Schwarz and triangle inequalities,
Pythagorean theorem, parallelogram equality:
(a) Use the general properties of an inner product (·, ·) on a vector space V, with induced
norm ||u|| = V(u, u), to show the Cauchy-Schwarz inequality: For any u, v E V,
|(u, v)| < ||u|| ||v|-
(CS)
[Hint: For eacht eR, we have (u+tv, u+tv) > 0 (justify this). Expand this inner prod-
uct to get an expression that is quadratic in t; show that the condition for this quadratic
to remain non-negative for all t is equivalent to (CS). Of course for the geometric dot
product, involving the cosine of an angle, (CS) can be shown very easily...]
(b) For any norm || · || induced by an inner product, as in (a) above, prove the triangle
inequality: For any u, v E V,
||u+ v|| < ||u|| + ||v||,
(A)
with equality iff u and v are orthogonal.
[Hint: Square both sides. Of course for || · | to be a norm it must satisfy (A); this result
shows that for any inner product, the quantity V(u, u) indeed defines a norm.]

(c) Prove the Pythagorean theorem for general inner product spaces:
||u + v||? = ||u||2 + ||v||²
iff u and v are orthogonal.
Generalize this result to show that if {V1, V2,..., Vk} is an orthogonal set and a1, a2, ..., ak
are scalars, then
k
lajl||v;||².
j=1
(d) If a norm || · || is induced by an inner product, as in (a) above, prove the parallelogram
law: For any u, v € V,
||u+ v||? + ||u – v||² = 2||u||2 + 2||v||?,
(P)
and motivate the name "parallelogram law" by giving an interpretation for geometric
vectors.

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