A10_soln - Math 235 Assignment 10 Solutions 1 Determine...

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Math 235 Assignment 10 Solutions 1. Determine whether the system is consistent, and if so, determine the general solution. (a) z 1 + (2 + i ) z 2 + iz 3 = 1 + i iz 1 + ( - 1 + 2 i ) z 2 + 2 iz 4 = - i z 1 + (2 + i ) z 2 + (1 + i ) z 3 + 2 iz 4 = 2 - i Solution: Row reducing gives 1 2 + i i 0 1 + i i - 1 + 2 i 0 2 i - i 1 2 + i 1 + i 2 i 2 - i R 2 - iR 1 R 3 - R 1 1 2 + i i 0 1 + i 0 0 1 2 i 1 - 2 i 0 0 1 2 i 1 - 2 i R 1 - iR 2 R 3 - R 2 1 2 + i 0 2 - 1 0 0 1 2 i 1 - 2 i 0 0 0 0 0 Hence, the system is consistent with two parameters. Let z 2 = s C and z 4 = t C . Then, the general solution is ~ z = - 1 0 1 - 2 i 0 + s - 2 - i 1 0 0 + t - 2 0 - 2 i 1 (b) iz 1 + 2 z 2 - (3 + i ) z 3 = 1 (1 + i ) z 1 + (2 - 2 i ) z 2 - 4 z 3 = i iz 1 + 2 z 2 - (3 + 3 i ) z 3 = 1 + 2 i Solution: Row reducing gives i 2 - 3 - i 1 1 + i 2 - 2 i - 4 i i 2 - 3 - 3 i 1 + 2 i R 2 - R 1 R 3 - R 1 i 2 - 3 - i 1 1 - 2 i - 1 + i - 1 + i 0 0 - 2 i 2 i R 1 - iR 2 1 2 iR 3 0 0 - 2 2 + i 1 - 2 i - 1 + i - 1 + i 0 0 1 - 1 R 1 + 2 R 3 0 0 0 i 1 - 2 i - 1 + i - 1 + i 0 0 1 - 1 Hence, the system is inconsistent.
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2 2. Find a basis for the four fundamental subspaces of A = 1 i 1 + i - 1 + i - 1 i . Solution: Row reducing the matrix
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A10_soln - Math 235 Assignment 10 Solutions 1 Determine...

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