istory Bookmarks Window Help ? d21 pds.edu subject) saad7@pdx.edu Portland State
ID: 3138510 • Letter: I
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istory Bookmarks Window Help ? d21 pds.edu subject) saad7@pdx.edu Portland State U Ind ?0.5 0 0 10. Let A 0.50 0 0 0.5 0 , and v b Define T:R3R3 by T(x)Ax. Find T(u) and T(v). In Exercises 11 and 12 with T defined by T(x) = Ax, find a vector x whose image under T is h, and determine whether x is unique. 1 3 2 3 -5-9 1-2 1 12. A4 5 -3 54 13. Find all x in R4 that are mapped into the zero vector by the transformation x Ax for the given matrix A. 1034 10 34 -2 3 05 14. Let b, and let A be the matrix in Exercise 13. Is b in the range of the linear transformation x Ax? Why or why not? 15, let T be defined by T(x) 1 01 1272 Use a rectangular coordinate system to plot u v, and their images under the given transformation T. Describe geometrically what T does to ? | 2 |12Explanation / Answer
11. Since det(A) ?0, hence A is invertible and A-1 =
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Let X be the vector such that T(X) = b. Then AX = b so that X = A-1b = A-1(6,-7,-9)T = (-5,-3,1)T. Further, X is unique.
12. A is a 4x3 matrix so we cannot compute its inverse. Let M = [A|b] =
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1
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9
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The RREF of M is
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0
If X = (x,y,z)T, then the equation AX = b is equivalent to x +3z= 7 or, x = 7-3z and y+z = 3 or, y = 3-z so that X = (7-3z,3-z,z)T = (7,3,0)T+t(-3, -1,1)T , where t = z is an arbitrary real number. Apparently, X is not unique.
Please post the remaining questions again separately.
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