Transcription of SOLUTION: ASSIGNMENT 5 - UCB Mathematics | Department …
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SOLUTION: ASSIGNMENT 5 - UCB Mathematics | Department …
SOLUTION: ASSIGNMENT the subset of P2given below a subspace? Find a basis if it is.{p(t) :p (1)=p(2)}(p is the derivative).S . LetS={p(t) :p (1)=p(2)}. For anyp1,p2 S,k1,k2 R, by(k1 p1+k2 p2) |t=1=k1p 1(1)+k2p 2(1)=k1p1(2)+k2p2(2)=(k1 p1+k2 p2)|t=2we checkedk1 p1+k2 p2 S. ThereforeSis a (t)=a+bt+ct2 S,p (1)=p(2) b+2c=a+2b+4c a+b+2c=0 (a,b,c)=( 2 , , )where , R. Thenp(t)= 2 + t+ t2= (t 1)+ (t2 2), {t 1,t2 2}. Butt 1,t2 2 Sare obviously independent. Therefore (t 1,t2 2) is a basis as in {p(t) :1 0p(t)dt=0}S . LetS={p(t) : 10p(t)dt=0}. For anyp1,p2 S,k1,k2 R, by1 0(k1 p1+k2 p2)dt=k11 0p1(t)dt+k21 0p2(t)dt=k10+k20=0we checkedk1 p1+k2 p2 S. ThereforeSis a (t)=a+bt+ct2 S,1 0p(t)dt=0 a+b2+c3=0 (a,b,c)=( 2 3, , )where , R. Thenp(t)= 2 3+ t+ t2= (t 12)+ (t2 13), {t 12,t2 13. Butt 12,t 13 Sare obviously independent. Therefore (t 12,t 13) is a basis a basis for the space of all diagonal2 2matrices, and determine its.}
Solution. Any diagonal n n matrix looks like 0 BBB BBB BBB B@ a1 an 1 CCC CCC CCC CA = a1E11 + + anEnn where Eii is the matrix with entries all 0 except a 1 at the i’th diagonal entry. This tells us that (E11; ;Enn)is a basis because these n matrices are already independent as in Rnn.The dimension is n. 4.1.24 Find a basis of the space of all upper triangular 3 3 matrices and determine its ...
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