# A10 - C n Prove that there exists vectors ~x,~ y ∈ R n...

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Math 235 Assignment 10 Due: Wednesday, July 20th 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 (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 2. Find a basis for the four fundamental subspaces of A = 1 i 1 + i - 1 + i - 1 i . 3. Diagonalize the following matrices over C . (a) A = 2 2 - 1 - 4 1 2 2 2 - 1 (b) A = 2 1 - 1 2 1 0 3 - 1 2 4. Let ~ z
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Unformatted text preview: C n . Prove that there exists vectors ~x,~ y ∈ R n such that ~ z = ~x + i~ y . 5. Let { ~v 1 ,~v 2 ,~v 3 } be a basis for R 3 . Prove that { ~v 1 ,~v 2 ,~v 3 } is also a basis for C 3 . 6. Prove that if ~ z is an eigenvector of a matrix A with complex entries, then ~ z is an eigenvector of A . What is the corresponding eigenvalue?...
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