MATH307 Group Homework9 Solved

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Instructions: Read textbook pages 93 to 100 before working on the homework problems. Show all steps to get full credits.

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1. Let e₁ = ,e₂ =    , V = span(e₁,e₂ − e₁),W = span(e₂) be two

0  1 subspaces of C², prove that C² = V + W but it is not a direct sum.

2. Let V = span{e₁,e₂},W = span{e₂,e₃}, where e₁,e₂,e₃ are vectors in R³, prove R³ = V + W. Is the sum a direct sum? Justify your answer.
3. Let A be a m × n complex valued matrix, use SVD to prove that C^m = range(A)⊕ null(A^∗).