1.8 EXERCISES 9 2 0 - 1. Let A = and define T : R2 R2 by T(x) = Ax. 10. A= 1 1 0 -2 3 0 1 3 که توا نا = [-] 2 0 Find the

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1.8 EXERCISES 9 2 0 - 1. Let A = and define T : R2 R2 by T(x) = Ax. 10. A= 1 1 0 -2 3 0 1 3 که توا نا = [-] 2 0 Find the images under T of u = 3 5 - [8] and y = a 11. Let b = , and let A be the matrix in the range of the linear transformation x not? .5 0 0 2. Let A= 0 .5 0 and y = 0 0 .5 Define TR → R by T(x) = Ax. Find T(u) and T(v). In Exercises 3-6, with 7 defined by T(x) = Ax, find a vector x whose image under 7 is b, and determine whether x is unique. 1 0-2 3. A= -2 b 3-2-5 12. Let b = 3 -1 , and let A be the matrix 1 6 --[] [:] b in the range of the linear transformation why not? In Exercises 13–16, use a rectangular coordina 4. A= 3 2 0 1 -4 3 - 5 - 9 b= u= -=[{]= [ ].and their images under V 5. A= b= 7 mation T. (Make a separate and reasonably lar exercise.) Describe geometrically what I does in R2 13. T(x) = 6. A= 1 3 0 -3 b = -4 1 5 1 -4 9 3 -6 14. 0 -[---][3] T®) = [ ][:] 15. Ta) = [: :][] T(x) = [1 ][:] 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T : RR by T(x) = Ax? 8. How many rows and columns must a matrix A have in order to define a mapping from R* into R$ by the rule T'(x) = Ax? For Exercises 9 and 10. find all x in R that are mapped into the zero vector by the transformation x - Ax for the given matrix A. -4 7 -5 9. A = 1 16. 17. Let T:R? → R2 be a linear transform u= into and maps v = iD 0 2 -6 -4 3 6 -4 fact that T is linear to find the images und 3u + 2v.