Let T R 3 R be linear Show that exist scalars a, b, and c such that T(x, y, z) ax by cz for all (x, y, z) R 3 Can you generalize this result for T F n F State and prove an analogous result for T F n F m

Question: Let T:R 3 R be linear. Show that exist scalars a, b, and c such that T(x, y, z) = ax + by +

Let T:R ³ → R be linear. Show that exist scalars a, b, and c such that T(x, y, z) = ax + by + cz for all (x, y, z) ∈ R ³. Can you generalize this result for T: F ⁿ → F? State and prove an analogous result for T: F ⁿ → F ^m.

Step by Step Solution

★★★★★

3.52 Rating (162 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

To address your question we should first demonstrate that if T mathbbR3 to mathbbR is a linear trans... 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 (2 attachments)

609a44a717257_30439.pdf

180 KBs PDF File

609a44a717257_30439.docx

120 KBs Word File

Students Have Also Explored These Related Mathematics Questions!

A company that manufactures three products a b and c requires two raw materials - labor and material - need to determine the optimal production that maximizes its total profit. A linear program was...

Determine the constants b and c such that the line tangent to f(x) = x 2 + bx + c at x = 1 is y = 4x + 2.

Amounts to be received by a partner during liquidation. A condensed balance sheet for a partnership to be liquidated is as follows: The profit and loss percentages for Partners A, B, and C are 50%,...

In a Hopfield neural network configured as an associative memory, with all of its weights trained and fixed, what three possible behaviours may occur over time in configuration space as the net...

3 Systems of differential equations Theorem 3. Any n linearly independent vectors in an n dimensional space V must also span V. That is to say, any n linearly independent vectors in an n dimensional...

Please answer all the questions. Thanks. Based on Exhibi 6-9 (Schedule A; Itemized deductions), Also attached: . 2017 Schedule A (same as Exhibit 6-9) 2018 Schedule A (draft) 2018 Schedule A...

Give examples of actions that would raise the GDP, including undesirable actions. BASED ON THIS ARTICLE "New Estimates of Over 500 Years of Historic GDP and Population Data" Fariss, C. J., Anders,...

MATHEMATICS FOR MACHINE LEARNING Marc Peter Deisenroth A. Aldo Faisal Cheng Soon Ong Contents Foreword 1 Part I Mathematical Foundations 9 1 Introduction and Motivation 11 1.1 Finding Words for...

chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS The purpose of this chapter is to introduce you to matrix algebra, which has many applications. You are already familiar with several algebras:...

A firm relies on R&D to maintain profitability. The firm needs to determine the maximum amount to invest today (or invest in phase I) for its three-phased project. Phase I: Research. (Invest R{)...

Find the derivative of Pt. Consider the combination of the temperature cycles (Subsection 1.8.3) given by Pa(t) 36.8+0.3 cos(2 P(t) 36.8+0.2 cos 0.583)) 2 16) (2m(28 16)) P, (t) = 36.8 + 0.2 cos +...

Determine whether the series is convergent or divergent. If it is converge n = 1 8 n 1 9 - n Need Help?

Given i(t)=5 cos(400t 120) A, determine the period of the current and the frequency in hertz.

Describe how to clone a LinkedBinaryTree instance representing a (not necessarily proper) binary tree, with use of the addLeft and addRight methods.

Venturi-type applicators are frequently used to spray liquid fertilizers. Water flowing through the venturi creates a sub atmospheric pressure at the throat, which in turn causes the liquid...

An invoice dated August 4 for $775.50 has terms 2/10, EOM. The customer received the shipment on July 30. How much would the customer need to pay on September 5 to fully settle the account?

The large power-cable reel is rolled up the incline by the vehicle as shown. The vehicle starts from rest with x = 0 for the reel and accelerates at the constant rate of 2 ft/sec 2 . For the instant...

WIf the general price level changes from 8% to 6%, then disinflation occurs. True Falsehen economic activity in the agricultural sector decreases due to a prolonged drought, this is an example of...

3. Let f: 3 - R be defined by g(r, y, z) = 2x2 + + 23. (a) Compute the gradient Vg. (b) The equation 2x2 + y2 + 23 = 14 determines a surface S in R3. Find the equation for the tangent plane to S at...

S Collectibles (coupon collectors problem). Suppose a company attaches to its product a sticker showing one of the players of the national football team. How many items do you have to buy on average...

I need help with this payroll project from Chapter 7. This is the book: Payroll Accounting, by Beig, Toland, edition 2017. I attached an excel template with the beginning filled out, and just need to...

Exercise 4.1 To develop some facility with this notation, let Tr = [ti j], where ti j =ir Nr n for r = 1, . . . ,t. Find T r Tr, T r Ts for s = r, and JTr. A very important application of this...

An event independently occurs on each day with probability p. Let N(n) denote the total number of events that occur on the first n days, and let Tr denote the day on which the rth event occurs. (a)...