Exercise 6.3.2 Which of the following subsets of V are independent?
a. V=P2;{x^2+1,x+1,x}
b. V=P2;{x^2-x+3,2x^2+x+5,x^2+5x+1}
c. V=M22;{[[1,1],[0,1]],[[1,0],[1,1]],[[1,0],[0,1]]}
d. V=M22;{[[-1,0],[0,-1]],[[1,-1],[-1,1]],[[1,1],[1,1]],[[0,-1],[-1,0]]}
e. V=F[1,2];{(1)/(x),(1)/(x^2),(1)/(x^3)}
f. V=F[0,1];{(1)/(x^2+x-6),(1)/(x^2-5x+6),(1)/(x^2-9)}
Exercise 6.3.3 Which of the following are independent in F[0,2Ï€]?
a. {sin^2x,cos^2x}
b. {1,sin^2x,cos^2x}
c. {x,sin^2x,cos^2x}
Exercise 6.3.1 Show that each of the following sets of vectors is independent.
d. V=M22; {[--],[-}-],[--}]
e. V=F[1,2];{,,} {-2+5-29-x+2}:[10]=J
Exercise 6.3.3 Which of the following are independent in F[0, 2Ï€]?
a. {sin2x,cos2x}
b. {1,sin2x,cos2x}
c. {x,sin2x,cos2x}
a. {1+x,1-x,x+x^2} in P
b. {x^2,x+1,1-x-x^2} in P2
{[][}8][1-] [81]} in M22
{[][i][}i] [1]} in M22
Exercise 6.3.4 Find all values of a such that the following are independent in R3.
a. {(1, -1, 0), (a, 1, 0), (0, 2, 3)}
b. {(2, a, 1), (1, 0, 1), (0, 1, 3)}
Exercise 6.3.5 Show that the following are bases of the space V indicated.
c. V=Mn{[i][Hi][i]}
a. {1,1,0),(1,0,1, 0,1,1};V=R3
