Let the linear map T : R2 R2 have standard matrix A = 2 1 1 2 _ . a. Calculate the

Chapter 4, Problem 27

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QUESTION:

Let the linear map T : R2 R2 have standard matrix A = 2 1 1 2 _ . a. Calculate the matrix for T with respect to the basis B = # 1 2 1 1 _ , 1 2 1 1 _$ . b. Set y = CB(x), and calculate Ax x = 2x2 1 2x1x2 + 2x2 2 in terms of y1 and y2. c. Use the result of part b to sketch the conic section 2x2 1 2x1x2 + 2x2 2 = 3. (See also Section 4.1 of Chapter 6.)

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QUESTION:

Let the linear map T : R2 R2 have standard matrix A = 2 1 1 2 _ . a. Calculate the matrix for T with respect to the basis B = # 1 2 1 1 _ , 1 2 1 1 _$ . b. Set y = CB(x), and calculate Ax x = 2x2 1 2x1x2 + 2x2 2 in terms of y1 and y2. c. Use the result of part b to sketch the conic section 2x2 1 2x1x2 + 2x2 2 = 3. (See also Section 4.1 of Chapter 6.)

ANSWER:

Step 1 of 5

a. Use the basis  to determine the matrix for

It is known that the standard matrix for  is .

Let  and .

Find the image of  under  and write it as a linear combination of  and .

So,

 

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