3.1. Let A Mn (F), and consider the linear transformation fA F F defined by...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
3.1. Let A € Mn (F), and consider the linear transformation fA F F defined by f₁(v) = Av. Suppose that there is a diagonal matrix D = 1 AjEjj and an invertible matrix P such that A = PDP-¹ Let vj denote the j-th column of P. Show that B = V1,..., Un is a basis of F consisting of eigenvectors for fA. Conclude that fa is diagonalizable. B 3.2. Let B = v₁,... Un be any basis of F. Write down the matrix [T¹], of T¹ L(F¹,F), where is the standard basis € = e₁,..., en of Fr. 3.3. Now let AE M (F) be any nxn matrix, and let B be as in Problem 3.2. Show that A = P[fa],P-1, where P is the invertible matrix with j-th column vj. Conclude that if f is diagonalizable, then the matrix A is diagonalizable as well (i.e. there is an invertible matrix P and a diagonal matrix D such that A = = PDP-¹). Hint. From Lecture 33, [fa], is equal to the matrix of (T) = TƒÃ° Twith respect to the standard basis C = ₁,..., en of F. Use this and that composition of linear transformations corresponds to matrix multiplication. O 3.1. Let A € Mn (F), and consider the linear transformation fA F F defined by f₁(v) = Av. Suppose that there is a diagonal matrix D = 1 AjEjj and an invertible matrix P such that A = PDP-¹ Let vj denote the j-th column of P. Show that B = V1,..., Un is a basis of F consisting of eigenvectors for fA. Conclude that fa is diagonalizable. B 3.2. Let B = v₁,... Un be any basis of F. Write down the matrix [T¹], of T¹ L(F¹,F), where is the standard basis € = e₁,..., en of Fr. 3.3. Now let AE M (F) be any nxn matrix, and let B be as in Problem 3.2. Show that A = P[fa],P-1, where P is the invertible matrix with j-th column vj. Conclude that if f is diagonalizable, then the matrix A is diagonalizable as well (i.e. there is an invertible matrix P and a diagonal matrix D such that A = = PDP-¹). Hint. From Lecture 33, [fa], is equal to the matrix of (T) = TƒÃ° Twith respect to the standard basis C = ₁,..., en of F. Use this and that composition of linear transformations corresponds to matrix multiplication. O
Expert Answer:
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
Students also viewed these mathematics questions
-
A Single Buyer in the Labor Market: The text treated extensively the case where market power is concentrated on the supply side but it could equally well be concentrated on the demand side. When a...
-
Show that there is no diagonal matrix similar to the matrix given in Example 4 of Section 9.1, 003 120 200
-
Woods Company has one depreciable asset valued at $800,000. Because of recent losses, the company has a net operating loss carry-forward of $150,000. The tax rate is 30%. The company was acquired for...
-
Show that 1. Every differentiable concave function is pseudoconcave. 2. Every pseudoconcave function is quasiconcave
-
A 6000-Hz tone must have what sound level to seem as loud as a 100-Hz tone that has a 50-dB sound level? (See Fig. 12-6)
-
A cylinder/piston contains 4 ft3 of air at 16 lbf/in 2, 77 F. The air is compressed in a reversible polytropic process to a final state of 120 lbf/in 2, 400 F. Assume the heat transfer is with the...
-
To calculate the expected frequencies, we must know the row totals, the column totals, and the ________________ total. In Exercises 35, fill in each blank with the appropriate word or phrase.
-
Calculating Discounted Payback An investment project costs $15,000 and has annual cash flows of $3,700 for six years. What is the discounted payback period if the discount rate is zero percent what...
-
Irvin Racquet Company produces tennis racquets. The direct labour standard for each racquet is 0.50 hours at a cost of $22.00 per direct labour hour. During the month of April, Irvin Racquet Company...
-
Using License Fees to Make Positive Profit: Suppose you own one of many identical pharmaceudical companies producing a particular drug x. A: Your production process has decreasing returns to scale...
-
Compute the thrust required for take-off for a 3-spool turbofan given the following data: a) Flight conditions: M = 0.2, T = 288 K, P = 1 bar b) cH = 5, = 0.98, T = 5, f = 0.0177, hp = 45000 kJ/kg...
-
McLovin is trying to make a fake ID using his highschool photocopier. Photocopiers work by using electric forces to get ink to bind to paper. Let's assume that each ink particle has a weight of one...
-
"Using the concept of "the rural", what is the impact of out-migration on rural communities?" I need to do an essay that demonstrates how the application of concepts and theories from rural sociology...
-
What is the RURAL AND URBAN SOCIOLOGY all about? How is it a relevant course?
-
a) You will use the sample wedding itinerary provided in the Assignment Part 1 attachment to MAKE a Wedding Planner Timeline for yourself as the Wedding Planner (does not need to look pretty or...
-
In this exercise, we assume that the following MIPS code is executed on a pipelined processor with a 5-stage pipeline, full forwarding, and a predict-taken branch predictor. LW Labell: BEQ LW R2, 0...
-
Explain the principle of chain surveying.
-
Suppose that the electrical potential at the point (x, y, z) is E(x, y, z) = x + y - 2z. What is the direction of the acceleration at the point (1,3,2)?
-
Dr. George E. Beeper is a single taxpayer. He lives at 45 Mountain View Dr., Apt. 321, Spokane, WA 99210. Dr. Beeper's Social Security number is 775-88-9531. Dr. Beeper works for the Pine Medical...
-
Brian and Kim have a 12-year-old child, Stan. For 2012, Brian and Kim have taxable income of $52,000, and Stan has nonqualifying dividend income of $4,500 and investment expenses of $250. No election...
-
Carl Conch and Mary Duval are married and file a joint return. They live at 1234 Mallory Sq. Apt. 64, Key West, FL 33040. Carl works for the Key Lime Pie Company and Mary is a homemaker after losing...
-
In the Hochfelder case all of the following were factors in the case except a. The Securities Exchange Act of 1934. Data From Exchange Act 1934 When the Securities Act was passed, the Interstate...
-
Refer to the example of an auditors report issued in 1915. List the differences between the report styles in 1915 and today (refer to Chapter 1). Indicate in what ways this report would be deficient...
-
What client actions might cause a CPA to terminate an existing client relationship?
Study smarter with the SolutionInn App