Let v and w be elements of an inner product space. Prove that ||v + w||2 =

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, w are orthogonal. Explain why this formula can be viewed as the generalization of the Pythagorean Theorem.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Applied Linear Algebra

ISBN: 978-0131473829

1st edition

Authors: Peter J. Olver, Cheri Shakiban

See More Books

Question Posted: Jun 17, 2016 08:47 AM

See More Questions

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.
As purchasing agent for Laurentian Enterprises in Quebec City, you ask your buyer to provide you with a ranking of excellent, good, fair or poor for a variety of characteristics for two potential...
Describe the concept of Six Sigma quality. Why is such a high quality level important?
What were the major organizational changes Lafley made at P&G, and what challenges still remain?
True or False: The technician should never need to check exhaust piping, equipment supports with torsional pivots, and so on, because shaft alignment is all that is necessary.
Nutritional Foods reports merchandise inventory at the lower-of-cost-or-market. Prior to releasing its financial statements for the year ended March 31, 2017, Nutritionals preliminary income...
HFA has two program departments, one for the treatment of ex - convicts who are drug users and the other for a job training program for ex - convicts. The treatment pro - gram hires 4 0 % of the...
Allie has bought a new apple orchard. The orchard has a single file of trees, numbered from 1 to N. Each tree has a certail number of ripe apples. Allie has a rule she wants to follow. She wants to...
Describe all orthonormal bases of R2 for the inner products (a) (b) 0 2 -1 2
Prove that if v1, v2 form a basis of an inner product space V and ||v1|| = ||v2|| then v1 + v2 and v1 - v2 form an orthogonal basis of V.
The accounting records of Faviana Shipping show the following assets and liabilities as of December 31, 2008, and 2009. Late in December 2009, the business purchased a small office building and land...
Find the APR (rounded to the nearest tenth of a percent) for each of the loans described in Problems 39-42. Purchase a refrigerator for $\$ 2,100$ at $14 \%$ add-on interest for 3 years.
In Problems 20-52: a. State the type; and $\mathbf{b}$. Answer the question. Mark's Grocery Store is going to be remodeled in 5 years, and the remodeling will cost $\$ 3,000,000$. How much should...
An insurance policy offers you the option of being paid $\$ 750$ per month for 20 years or a lump sum of $\$ 100,000$. Which has more value if the current rate of return is $5.5 \%$, compounded...
a. Find the first three terms of the sequences whose nth terms are given. b. Classify the sequence as arithmetic (give d), geometric (give r), both, or neither. $s_{n}=\frac{n-1}{n+1}$
Round your answers in Problems 35-38 to the nearest dollar. Make a $10 \%$ offer on a Saturn that has a sticker price of $\$ 19,895$ and a dealer cost of $\$ 17,250$.
Revenue and expense accounts are listed in the Trial Balance section of the work sheet and in the ________ of the work sheet. a. Retained Earnings statement b. Balance Sheet section c. Income...
Prove the result that the R 2 associated with a restricted least squares estimator is never larger than that associated with the unrestricted least squares estimator. Conclude that imposing...
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.
A) Translate the following C into RV641 assembly. Be sure to follow the calling (Refer to slide22 of lec6_postZoom.pdf) and comment each line of your assembly. Remember that long long is 64 bits in...
Given a Binary Tree, find the maximum sum path from a leaf to root. Input: 1 A Input: 10 -27 4 23 Output: Explanation Following the path 3-1, results in a sum of 4, which is the maximum path sum from...
On December 1 , 2 0 1 5 , your company pays $ 2 4 , 0 0 0 for rent on your store that covers the next 6 months. When you paid the cash, you debited Prepaid Rent.. The adjusting journal entry on...

Let v and w be elements of an inner product space. Prove that ||v + w||2 =

Question:

Step by Step Answer:

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

Applied Linear Algebra

Students also viewed these Linear Algebra questions