4. For each of the following linear transformations, express it in (5.1.1) O. a. T(x1, x2)...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
4. For each of the following linear transformations, express it in (5.1.1) O. a. T(x1, x2) = (x1 – x2, 2x1 +x2, 5x1 – 3x2). Definition 5.1.1. Let A be an m x n matrix given in (2.1.1)O. A linear transformation from R" to R". denoted by T, is defined by (5.1.1) a11 a12 din ... a21 a22 a2n ... T(X)= AX = %3D Aml amn In ... where X = (*1, L2,- ,n)") E R". The vector T(X ) is said to be the image of X under Tand X is called the inverse image of T(X ). The space R" is the domain of Tand R" is the codomain of T. The matrix A is said to be the standard matrix for Tand Tis called the multiplication by A. A linear transformation from R" to R" is said to be a linear operator. If Tis a linear transformation from R" to R", then we say that T maps R" into R", which is denoted by the symbol T: R" → R" When m =n = 1, the linear transformation is a function T:R -R defined by T(x) = a1r. Hence, linear transformations are generalizations of such functions to higher dimensional spaces. By Definition 5.1.1 0, we see that every matrix determines a linear transformation. Hence, for every linear transformation, there is a unique matrix corresponding to it. Sometimes, we write T as T, in order to emphasize that the linear transformation is determined by the standard matrix A. 4. For each of the following linear transformations, express it in (5.1.1) O. a. T(x1, x2) = (x1 – x2, 2x1 +x2, 5x1 – 3x2). Definition 5.1.1. Let A be an m x n matrix given in (2.1.1)O. A linear transformation from R" to R". denoted by T, is defined by (5.1.1) a11 a12 din ... a21 a22 a2n ... T(X)= AX = %3D Aml amn In ... where X = (*1, L2,- ,n)") E R". The vector T(X ) is said to be the image of X under Tand X is called the inverse image of T(X ). The space R" is the domain of Tand R" is the codomain of T. The matrix A is said to be the standard matrix for Tand Tis called the multiplication by A. A linear transformation from R" to R" is said to be a linear operator. If Tis a linear transformation from R" to R", then we say that T maps R" into R", which is denoted by the symbol T: R" → R" When m =n = 1, the linear transformation is a function T:R -R defined by T(x) = a1r. Hence, linear transformations are generalizations of such functions to higher dimensional spaces. By Definition 5.1.1 0, we see that every matrix determines a linear transformation. Hence, for every linear transformation, there is a unique matrix corresponding to it. Sometimes, we write T as T, in order to emphasize that the linear transformation is determined by the standard matrix A.
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
For each of the following linear transformations L: R2 R2, find a matrix representative, and then describe its effect on (i) The x-axis; (ii) The unit square S = (0 x, v 1): (iii) The unit disk D...
-
Let A be an m x n matrix with rank A = n. (a) Prove that the matrix P = A(ATA)~l AT is a projection matrix, meaning that P2 = P, cf. Exercise 2.5.8. (b) Construct the projection matrix corresponding...
-
Let A be an m x n matrix whose column vectors are mutually orthogonal, and let b Rm. Show that if y is the least squares solution to the system Ax = b then b' a
-
Rotorua Products, Ltd., of New Zealand markets agricultural products for the burgeoning Asian consumer market. The company's current assets, current liabilities, and sales over the last five years...
-
What kinds of data can be gathered through electronic scanner services?
-
Brantley Shumaker opened a public relations firm called Gold Star Services on August 1, 2025. The following amounts summarize the business on August 31, 2025: During September 2025, the business...
-
With reference to Exercise 11.65, construct a \(99 \%\) confidence interval for \(\alpha\).
-
McMinville Corporation manufactures paper products. In 2007, the company purchased several large tracts of timber for $22 million with the intention of harvesting the timber rather than buying this...
-
The Changing Workforce The composition of the modern labor force is changing rapidly. Increasing diversity in race, ethnicity, and gender, a shift in age distribution, and a shift in the skill levels...
-
Maria Jones has just earned a university degree in management. She has taken the position of assistant to the president at Roseville Engineering, which manufacturers tungsten carbide drill steels for...
-
Perry Mason asked his secretary Della Street to take her car and deliver a pleading to the courthouse for filing. Driving to the courthouse, Della accidentally ran a stop sign and caused a driver to...
-
Explain how customer lifetime value analysis can be applied to improve all aspects of the customer lifecycle.
-
Corporate standards for a websites format and update process are likely to stifle the creative development of a site and reduce its value to customers. Discuss.
-
Choose a service company you are familiar with, and show how each of the 7 Ps of services marketing applies to one of its service products.
-
How do the concepts of theater, role theory, and script theory help to provide insights into consumer behavior during the service encounter?
-
What are the key elements in the framework for developing effective service marketing strategies?
-
Based on the below analysis, which portfolio would you suggest to Sandy? Why? Risk-free rate (rf) = 3% Portfolio 1: 100% allocation to Acme Company expected return: E(rp)=w1 x E(r1) + Wrfx Rrf=...
-
What types of inventory issues Starbucks might reflect upon at the end of each year? The mission of Starbucks is to inspire and nurture the human spiritone person, one cup, and one neighborhood at a...
-
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...
-
Use Householder matrices to convert the following matrices into upper Hessenberg form: (a) (b) (c) 3-2 1025 2410 3201 1134 0101 1213
-
Solve the following linear systems by the conjugate gradient method, keeping track of the residual vectors and solution approximations as you iterate. (a) (b) (c) (d) (e) 102 112 62 -3 4 9 1201 LI...
-
Describe how partial and circular reasoning can be helpful or harmful in resolving ethical dilemmas.
-
Describe the various legal risks for nurses.
-
Describe the various roles of advanced practice nurses.
Study smarter with the SolutionInn App