MATH307 Individual Homework5 Solved

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Instructions: Read textbook pages 31 to 34 before working on the homework problems. Show all steps to get full credits.

1. Let P⁴ be the set of all real coefficient polynomials of degree less than or equal to 4, check whether each one of the following is a basis of P⁴ and justify your answer using exchange theorem:
   • {1,x,−x²,x³}.
   • {1,1+ x,1+ x + x²,x² + x³,x³ − x⁴}.
   • {−x⁴,x³,−x²,x,−1}.
   • {5,x⁴,x³ − x²,x² − x,x +10,x² −5}.
2. Textbook page 40, Chapter 3 problem 6.
3. Textbook page 40, Chapter 3 problem 7.
4. What is the dimension of C^3×2 over C? Let e_ij a 3×2 matrix with ij-th entry equals to 1 and 0 elsewhere. Is e₁₁,e₁₂,e₂₁,e₂₂,e₃₁,e₃₂,e₃₂ − e₁₁ a basis of C^3×2? Justify your answer.