[Solved] MATH307 Individual Homework5

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 C³² over C? Let e_ij a 32 matrix with ij-th entry equals to 1 and 0 elsewhere. Is e₁₁,e₁₂,e₂₁,e₂₂,e₃₁,e₃₂,e₃₂ e₁₁ a basis of C³²? Justify your answer.