Let xTAx be a quadratic form in x1, x2,..., xn and define T: Rn R by

Question:

Let xTAx be a quadratic form in x1, x2,..., xn and define T: Rn → R by T(x) = xTAx.
(a) Show that T(x + y) = T(x) + 2xTAy + T(y).
(b) Show that T = (kx) = k2T(x).
(c) Is T a linear transformation? Explain.

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

Step by Step Answer:

Related Book For book-img-for-question

Elementary Linear Algebra with Applications

ISBN: 978-0471669593

9th edition

Authors: Howard Anton, Chris Rorres

See More Books

Question Posted: Jun 17, 2016 08:14 AM

See More Questions

Let the matrix A = (aij) represent a linear operator f with respect to the orthonormal basis x1; x2; . . . ; xn. Then the sum defines a quadratic form on X, where x1; x2; . . . ; xn are the...
For any quadratic form Q(x). xTAx, there exists a basis x1; x2; . . . , xn and numbers such that Q(x).
Let f be defined on [a, b], and let the nodes a = x0 < x1 < x2 = b be given. A quadratic spline interpolating function S consists of the quadratic polynomial S0(x) = a0 + b0(x x0) + c0(x x0)2 on...
If you were in charge of B&D, indicate the possible decisions you might take as regards the four Ps and the points you would want to keep in mind as you go about the task of marketing for...
A New Yorker travels to New Jersey to buy a $100 telephone answering machine. The New Jersey Company that sells the machine then deposits the $100 check in its account at a New York bank. How would...
Financial information is presented below for three different companies. Instructions Determine the missing amounts. Hardy Yee Wang Cosmetics Wholesalers Grocery $ (e) 5,000 95,000 (f) 38,000 (g)...
Culture and Mental Illness A recent study examining the link between schizophrenia and culture interviewed 60 people who had been diagnosed with schizophrenia and who heard voices in their heads. The...
Maverick Wings, Inc. manufactures airplanes for use in stunt shows. Maverick's factory is highly automated, using the latest in robotic technology. To keep costs low, the company employs as few...
Service revenue Municipal bond interest income Interest income from savings accounts Non-business income Total Income e Salaries and wages expense Rent expense Depreciation expense 5 Supplies expense...
Project Alpha requires an immediate investment of $16,000 and an additional investment of $12,000 in three years. Net returns are estimated to be $8,000 after two years, $24,000 after four years, and...
In each part, classify the quadratic form as positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite. (a) x21 + x22 (b) (x1 - x2)2 (c) x21 - x22
Express the quadratic form (c1x1 + c2x2 + ... + cnxn)2 in the matrix notation xTAx, where A is symmetric.
The economy of Britannica produces three goods: computers, pens, and pizza. The accompanying table shows the prices and output of the three goods for the years 2014, 2015, and 2016. a. What is the...
Describe a game you could use in your early childhood class to facilitate the development of articulation skills. Discuss how you would use this game and how it encourages the development of...
Francine is a 70-year-old, Irish Catholic female. She worked for 40 years as a librarian in an institution of higher education and retired at age 65. Francine has lived alone for the past year, after...
The term used by personality psychologists to describe the uniqueness of the individual. The description of personality involves judgments regarding who the person truly is and how she/he differs...
Categorize the accounting tasks performed by Callison as relating to either the identification ( I ) , recording ( R ) , or communication ( C ) aspects of accounting. Identify users of accounting...
You are the new Coding Manager. Locate the current Office of Inspector General Work Plan Active Issues on the Health and Human Services Office of Inspector General website. Review these Active Work...
An absolute upper bound on G E for stability of an equimolar binary mixture is G E = RT ln 2 . Develop this result. What is the corresponding bound for an equimolar mixture containing N species?
Write a paper about the Working relationship in the organization- collaboration within and outside the organization
For various values of k, form an k k matrix A by setting A = 2 * eye(k) - diag(ones(k - 1, 1), 1) - diag(ones(k - 1, 1), -1) In each case, compute the LU factorization of A and the determinant of A....
For any positive integer n, the MATLAB command P = pascal(n) will generate an n Ã n matrix P whose entries are given by The name pascal refers to Pascal's triangle, a triangular array of...
The fourth matrix has 0's on the diagonal and l's in the off-diagonal positions. Rotate the vector x around the unit circle and note when x and Ax are parallel. Based on these observations, determine...
Accounting for Fair Value Hedge: Interest Rate Swap On January 1 of Year 1, Innovative Lab issued a 4-year $50,000 note to a local bank with fixed interest payments based on 6%, payable annually on...
Part 4 Tennis company, a distributor of Tennis Rackets, has been experiencing losses for some time, as shown by its most recent monthly contribution format income statement below: Sales Variable...
[The following information applies to the questions displayed below.] Bunnell Corporation is a manufacturer that uses job-order costing. On January 1, the company's inventory balances were as...

Let xTAx be a quadratic form in x1, x2,..., xn and define T: Rn R by

Question:

Step by Step Answer:

a By definition Tx y x y T Ax y x T y T Ax y x T Ax x T Ay y T Ax y T A y Tx x T A...View the full answer

Elementary Linear Algebra with Applications

Students also viewed these Linear Algebra questions