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3. For this question you should use the fact (given on Assn.4) that the matrix -3 6 6 4-5 1 4 2 8-9 7 A = has reduced row-echelon form R= 0 233 3 9 -5 -9 -8 8 9 -12 6 15 10 3 20-24 01 -2 -2 0 7 00 0 0 1 4 0 0 0 000 a) Find a basis for the kernel and range of the linear transformation T: R6 → R¹ given by T(a, b, c, d, e, f) = (-3b+6c+6d+4e-5f, 2a+b+4c+2d+8e-9f, 3a+7b-5c-8d+8e+9f, 3a+9b-9c-12d+6e+15f) (HINT: this is easy - see Assn.4 for the relevant calculations) b) ■] Find a basis for the kernel and range of the linear transformation T : M2×3 → P3 given by b c T (a & ;). = (−3b +6c+ 6d+ 4e − 5ƒ)x³ d e +(2a+b+4c+2d+8e-9f)x²+(3a+7b-5c-8d+8e+9f)x+3a+9b-9c-12d+6e+15f

3. For this question you should use the fact (given on Assn.4) that the matrix -3 6 6 4-5 1 4 2 8-9 7 A = has reduced row-echelon form R= 0 233 3 9 -5 -9 -8 8 9 -12 6 15 10 3 20-24 01 -2 -2 0 7 00 0 0 1 4 0 0 0 000 a) Find a basis for the kernel and range of the linear transformation T: R6 → R¹ given by T(a, b, c, d, e, f) = (-3b+6c+6d+4e-5f, 2a+b+4c+2d+8e-9f, 3a+7b-5c-8d+8e+9f, 3a+9b-9c-12d+6e+15f) (HINT: this is easy - see Assn.4 for the relevant calculations) b) ■] Find a basis for the kernel and range of the linear transformation T : M2×3 → P3 given by b c T (a & ;). = (−3b +6c+ 6d+ 4e − 5ƒ)x³ d e +(2a+b+4c+2d+8e-9f)x²+(3a+7b-5c-8d+8e+9f)x+3a+9b-9c-12d+6e+15f

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. For this question you should use the fact (given on Assn.4) that the matrix has reduced row-echelon form A = (HINT: this is easy R= 0 2 3 3 T -3 6 1 4 7 -5 9 -9 -51 (a) Find a basis for the kernel and range of the linear transformation T: R6 → R4 given by T(a, b, c, d, e, f) = 2 8-9 -8 8 9 -12 6 15 1 0 3 2 0-241 01 -2 -2 0 7 00 0 1 000000 0 4 (-3b+6c+6d+4e-5f, 2a+b+4c+2d+8e-9f, 3a+7b-5c-8d+8e+9f, 3a+9b-9c-12d+6e+15f) - see Assn.4 for the relevant calculations) b (² (a ! ;) d e f (b) Find a basis for the kernel and range of the linear transformation T: M2×3 → P3 given by = = (−3b +6c+ 6d+ 4e - 5f)x³ +(2a+b+4c+2d+8e-9f)x²+(3a+7b-5c-8d+8e+9f)x+3a+9b-9c-12d+6e+15f Make sure the basis vectors are in the correct form (matrices or polynomials).
Transcribed Image Text:3. For this question you should use the fact (given on Assn.4) that the matrix has reduced row-echelon form A = (HINT: this is easy R= 0 2 3 3 T -3 6 1 4 7 -5 9 -9 -51 (a) Find a basis for the kernel and range of the linear transformation T: R6 → R4 given by T(a, b, c, d, e, f) = 2 8-9 -8 8 9 -12 6 15 1 0 3 2 0-241 01 -2 -2 0 7 00 0 1 000000 0 4 (-3b+6c+6d+4e-5f, 2a+b+4c+2d+8e-9f, 3a+7b-5c-8d+8e+9f, 3a+9b-9c-12d+6e+15f) - see Assn.4 for the relevant calculations) b (² (a ! ;) d e f (b) Find a basis for the kernel and range of the linear transformation T: M2×3 → P3 given by = = (−3b +6c+ 6d+ 4e - 5f)x³ +(2a+b+4c+2d+8e-9f)x²+(3a+7b-5c-8d+8e+9f)x+3a+9b-9c-12d+6e+15f Make sure the basis vectors are in the correct form (matrices or polynomials).
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