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    Modify Figure 3.1 to illustrate the four fundamental subspaces asso-ciated withATIRnmthought of as a transformation from IRmtoIRn.N(AT)(35.000,105.000)(30.156,102.344)(25.625,99.375)(21.406,96.094) (17.500,92.500)(13.906,88.594)(10.625,84.375)(7.656,79.844) (5.000,75.000) (5.000,75.000)(3.828,72.500)(2.812,70.000)(1.953,67.500) (1.250,65.000)(0.703,62.500)(0.312,60.000)(0.

    078,57.500) (0.000,55.000)(0.078,52.500)(0.312,50.000)(0.703,47.500) (1.250,45.000)(1.953,42.500)(2.812,40.000)(3.828,37.500) (5.000,35.000) (5.000,35.000)(7.656,30.156)(10.625,25.625)(13.906,21.406) (17.500,17.500)(21.406,13.906)(25.625,10.625)(30.156,7.656) (35.000,5.000) (35.000,5.000)(37.500,3.828)(40.000,2.812)(42.500,1.953) (45.000,1.250)(47.500,0.703)(50.000,0.312)(52.500,0.078) (55.000,0.000)(57.500,0.078)(60.000,0.312)(62.500,0.703) (65.000,1.250)(67.500,1.953)(70.000,2.812)(72.500,3.828) (75.000,5.000) (75.000,5.000)(79.844,7.656)(84.375,10.625)(88.594,13.906) (92.500,17.500)(96.094,21.406)(99.375,25.625)(102.344,30.156) (105.000,35.000) (105.000,35.000)(106.211,37.500)(107.344,40.000)(108.398,42.500)(109.375,45.000)(110.273,47.500)(111.094,50.000)(111.836,52.500) (112.500,55.000)(113.086,57.500)(113.594,60.000)(114.023,62.500) (114.375,65.000)(114.648,67.500)(114.844,70.000)(114.961,72.500) (115.000,75.000) (115.000,75.000)(114.688,79.844)(113.750,84.375)(112.188,88.594) (110.000,92.500)(107.188,96.094)(103.750,9

    9.375)(99.688,102.344) (95.000,105.000) (95.000,105.000)(90.156,107.344)(85.625,109.375)(81.406,111.094) (77.500,112.500)(73.906,113.594)(70.625,114.375)(67.656,114.844) (65.000,115.000) (65.000,115.000)(60.625,115.000)(57.500,115.000)(55.000,115.000) (55.000,115.000)(53.750,114.375)(50.000,112.500)(47.188,111.094) (43.750,109.375)(39.688,107.344)(35.000,105.000)rR(AT)(35.000,105.000)(30.156,102.344)(25.625,99.375)(21.406,96.094) (17.500,92.500)(13.906,88.594)(10.625,84.375)(7.656,79.844) (5.000,75.000) (5.000,75.000)(3.828,7

    2.500)(2.812,70.000)(1.953,67.500) (1.250,65.000)(0.703,62.500)(0.312,60.000)(0.078,57.500) (0.000,55.000)(0.078,52.500)(0.312,50.000)(0.703,47.500) (1.250,45.000)(1.953,42.500)(2.812,40.000)(3.828,37.500) (5.000,35.000) (5.000,35.000)(7.656,30.156)(10.625,25.625)(13.906,21.406) (17.500,17.500)(21.406,13.906)(25.625,10.625)(30.156,7.656) (35.000,5.000) (35.000,5.000)(37.500,3.828)(40.000,2.812)(42.500,1.953) (45.000,1.250)(47.500,0.703)(50.000,0.312)(52.500,0.078) (55.000,0.000)(57.500,0.078)(60.000,0.312)(62.500,0.703) (65.000,1.250)(67.500,1.953)(70.000,2.812)(72.500,3.828) (75.000,5.000) (75.000,5.000)(79.844,7.656)(84.375,10.625)(88.594,13.906) (92.500,17.500)(96.094,21.406)(99.375,25.625)(102.344,30.156) (105.000,35.000) (105.000,35.000)(106.211,37.500)(107.344,40.000)(108.398,42.500)

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    (109.375,45.000)(110.273,47.500)(111.094,50.000)(111.836,52.500) (112.500,55.000)(113.086,57.500)(113.594,60.000)(114.023,62.500) (114.375,65.000)(114.648,67.500)(114.844,70.000)(114.961,72.500) (115.000,75.000) (115.000,75.000)(114.688,79.844)(113.750,84.375)(112.188,88.594) (110.000,92.500)(107.188,96.094)(103.750,99.375)(99.688,102.344) (95.000,105.000) (95.000,105.000)(90.156,107.344)(85.625,109.375)(81.406,111.094) (77.500,112.500)(73.906,113.594)(70.625,114.375)(67.656,114.844) (65.000,115.000) (65.000,115.000)(60.625,115.000)(57.500,115.000)(55.000,115.000) (55.000,115.000)(53.750,114.375)(50.000,112.500)(47.188,111.094) (43.750,109.375)(39.688,107.344)(35.000,105.000)rN(AT)(35.000,105.000)(30.156,102.344)(25.625,99.375)(21.406,96.094) (17.500,92.500)(13.906,88.594)(10.625,84.375)(7.656,79.844) (5.000,75.000) (5.000,75.000)(3.828,72.500)(2.812,70.000)(1.953,67.500) (1.250,65.000)(0.703,62.500)(0.312,60.000)(0.078,57.500) (0.000,55.000)(0.078,52.500)(0.312,50.000)(0.703,47.500) (1.250,45.000)(1.953,42.500)(2.812,40.000)(3.828,37.500) (5.000,35.000) (5.000,35.000)(7.656,30.156)(10.625,25.625)(13.906,21.406) (17.500,17.500)(21.406,13.906)(25.625,10.625)(30.156,7.656) (35.000,5.000) (35.000,5.000)(37.500,3.828)(40.000,2.812)(42.500,1.953) (45.000,1.250)(47.500,0.703)(50.000,0.312)(52.500,0.078) (55.000,0.000)(57.500,0.078)(60.000,0.312)(62.500,0.703) (65.000,1.250)(67.500,1.953)(70.00

    0,2.812)(72.500,3.828) (75.000,5.000) (75.000,5.000)(79.844,7.656)(84.375,10.625)(88.594,13.906) (92.500,17.500)(96.094,21.406)(99.375,25.625)(102.344,30.156) (105.000,35.000) (105.000,35.000)(106.211,37.500)(107.344,40.000)(108.398,42.500)(109.375,45.000)(110.273,47.500)(111.094,50.000)(111.836,52.500) (112.500,55.000)(113.086,57.500)(113.594,60.000)(114.023,62.500) (114.375,65.000)(114.648,67.500)(114.844,70.000)(114.961,72.500) (115.000,75.000) (115.000,75.000)(114.688,79.844)(113.750,84.375)(112.188,88.594) (110.000,92.500)(107.188,96.094)(103.750,99.375)(99.688,102.344) (95.000,105.000) (95.000,105.000)(90.156,107.344)(85.625,109.375)(81.406,111.094) (77.500,112.500)(73.906,113.594)(70.625,114.375)(67.656,114.844) (65.000,115.000) (65.000,115.000)(60.625,115.000)(57.500,115.000)(55.000,115.000) (55.000,115.000)(53.750,114.375)(50.000,112.500)(47.188,111.094) (43.750,109.375)(39.688,107.344)(35.000,105.000)m

    rR(AT)(35.000,105.000)(30.156,102.344)(25.625,99.375)(21.406,96.094) (17.500,92.500)(13.906,88.594)(10.625,84.375)(7.656,79.844) (5.000,75.000) (5.000,75.000)(3.828,72.500)(2.812,70.000)(1.953,67.500) (1.250,65.000)(0.703,62.500)(0.312,60.000)(0.078,57.500) (0.000,55.000)(0.078,52.500)(0.312,50.000)(0.703,47.500) (1.250,45.000)(1.953,42.500)(2.812,40.000)(3.828,37.500) (5.000,35.000) (5.000,35.000)(7.65

    6,30.156)(10.625,25.625)(13.906,21.406) (17.500,17.500)(21.406,13.906)(25.625,10.625)(30.156,7.656) (35.000,5.000) (35.000,5.000)(37.500,3.828)(40.000,2.812)(42.500,1.953) (45.000,1.250)(47.500,0.703)(50.000,0.312)(52.500,0.078) (55.000,0.000)(57.500,0.078)(60.000,0.312)(62.500,0.703) (65.000,1.250)(67.500,1.953)(70.000,2.812)(72.500,3.828) (75.000,5.000) (75.000,5.000)(79.844,7.656)(84.375,10.625)(88.594,13.906) (92.500,17.500)(96.094,21.406)(99.375,25.625)(102.344,30.156) (105.000,35.000) (105.000,35.000)(106.211,37.500)(107.344,40.000)(108.398,42.500)(109.375,45.000)(110.273,47.500)(111.094,50.000)(111.836,52.500) (112.500,55.000)(113.086,57.500)(113.594,60.000)(114.023,62.500) (114.375,65.000)(114.648,67.500)(114.844,70.000)(114.961,72.500) (115.000,75.000) (115.000,75.000)(114.688,79

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    .844)(113.750,84.375)(112.188,88.594) (110.000,92.500)(107.188,96.094)(103.750,99.375)(99.688,102.344) (95.000,105.000) (95.000,105.000)(90.156,107.344)(85.625,109.375)(81.406,111.094) (77.500,112.500)(73.906,113.594)(70.625,114.375)(67.656,114.844) (65.000,115.000) (65.000,115.000)(60.625,115.000)(57.500,115.000)(55.000,115.000) (55.000,115.000)(53.750,114.375)(50.000,112.500)(47.188,111.094) (43.750,109.375)(39.688,107.344)(35.000,105.000)nr{0}{0}ATAT(170,110)(255,150)!(170,260)(220,260)


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