Question: Let v and w be elements of an inner product space. Prove that ||v + w||2 = ||v||2 + ||w||2 if and only if v,

Let v and w be elements of an inner product space. Prove that ||v + w||2 = ||v||2 + ||w||2 if and only if v, w are orthogonal. Explain why this formula can be viewed as the generalization of the Pythagorean Theorem.

Step by Step Solution

★★★★★

3.36 Rating (165 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

v w 2 v w v w v v 2 v w w w ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

952-M-L-A-E (2195).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Problem 10-34 (LO. 6, 8) Ramon had AGI of $180,000 in 2023. He is considering making a charitable contribution this year to the American Heart Association, a qualified charitable organization....

Let v denote a vector in an inner product space V. (a) Show that W = (w | w in V,(v, w) = 0} is a subspace of V. (b) If V= R3 with the dot product, and if v = (1, -1, 2), find a basis for IV.

Let V be an inner product space. Show that if w is orthogonal to each of the vectors u1,u2, ... ,ur, then it is orthogonal to every vector in span (u1, u2,..., ur,}.

Let (v1, v2, v3) be an orthonormal basis for an inner product space V. Show that if is a vector in V, then ||w||2 = (w, v1)2 + (w, v2)2 + (w, v3)2.

Let v and w be vectors in an inner product space V. Show that: (a) v is orthogonal to w if and only if ||v + w|| = ||v-w||. (b) v + w and v - w are orthogonal if and only if ||v|| - ||w||.

How can I prove this problem? Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V. Then we can define a set X as X = {u + we Vu EU, WEW} Note that the...

Let M and N be subspaces of an inner product space V . We sa~.-r that the space V is the orthogonal direct sum of M and M if M + N = 'U' and M J. N. In this case we writeV= M 63 H. Prove that if an...

this question has already dued? i just want to know how to do 03 (6 points) Let V be an n-dimensional vector space with inner product ( , ), and let W be a subspace of V. Prove that for all 5 E \\V,...

Let Let (V, ( , )v) and (W, ( , )w) be finite-dimensional inner product spaces. Recall that the adjoint L* : W - V of a linear function L E Hom(V, W) is completely determined by the equation (L(v),...

(a) Let u, v, w be vectors in an inner product space (V, (., .) ). Show that if u and w are not orthogonal to each other, then there is a unique a E R such that u is orthogonal to v + aw. (b) Define...

How can I prove this? Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such that Unw ={0}. Prove that ( Ue W) + cul

Do you think offering incentives to teachers motivates them to do a better job in the classroom? Why or why not? Explain.

In general, this chapter has considered forces that are applied to objects of constant mass. What would be the situation if mass were added to or lost from a system while a constant force was being...

The oxygen content t days after organic waste was dumped into a pond is given by f ( t ) = 1 0 0 t 2 + 1 0 t + 1 0 0 t 2 + 2 0 t + 1 0 0 ( a ) Use the intermediate value theorem ( read and understand...

The life in hours of a 75-watt light bulb is known to be normally distributed with = 25 hours. A random sample of 20 bulbs has a mean life of X = 1014 hours. (a) Construct a 95% two-sided confidence...

The angular acceleration of a shaft is defined by the relation = 0.5, where is expressed in rad/s2 and in rad/s. Knowing that at t = 0 the angular velocity of the shaft is 30 rad/s, determine (a)...

A tennis player receives a shot with the ball (0.060 0 kg) traveling horizontally at 50.0 m/s and returns the shot with the ball traveling horizontally at 40.0 m/s in the opposite direction. (a) What...

A screen is placed 50.0 cm from a single slit, which is illuminated with 690-nm light. If the distance between the first and third minima in the diffraction pattern is 3.00 mm, what is the width of...

Two radio antennas separated by 300 m as shown in Figure P37.3 simultaneously broadcast identical signals at the same wavelength. A radio in a car traveling due north receives the signals. (a) If the...

Identify several occupations in which conflict resolution skills are particularly important.

In each part, find the principal argument of z. (a) z = l (b) z = - i (c) z = - 1 + 3i

In each part, find Re (z) and Im (z). (a) z = 3ei (b) |= 2ei/2|

Show that Formula 7 is valid if n = 0 or n is a negative integer.

Karen and Mike currently insure their cars with separate companies, paying $600 and $800 a year. If they insured both cars with the same company, they would save 10 percent on the annual premiums....

Need help with screenshots 1 through 8 (one through 8). Note: Screenshot 6 (six) includes two photographs because I could not fit both the prompt and graphs in one screenshot. Please number your...

please help me with ratio analysis please provide excel spreadsheet please Income Statement Spreadsheet Example Name: Jaspreet kaur Date: Jacob Corporation INTRODUCTION The purpose of financial...