Suppose al = (1, 0, 1), a2 = (0, 1, -2) and a3 = (-1, -1, 0) are vectors in R3 and f : R3 -> R is a linear functional such that f(al) = 1, f(a2) = -1 and f(a3) = 3.
If a = (p,q,r) belongs to R3, find f(a).
1
Expert's answer
2020-11-09T15:45:24-0500
Since ∣∣10−101−11−20∣∣=0+0+0−(−1)−0−(−1)(−2)=−1=0 , the vectors a1,a2,a3 are linearly independent in R3 , and thus form a basis of R3 . Therefore, each vector a=(p,q,r)∈R3 can be represent as a linear combination of a1,a2,a3, that is a=ta1+sa2+ma3 for some t,s,m∈R. Therefore, we have
(p,q,r)=t(1,0,1)+s(0,1,−2)+m(−1,−1,0)=(t−m,s−m,t−2s). Then for each (p,q,r) let us solve the following system:
⎩⎨⎧p=t−mq=s−mr=t−2s
which is equivalent to
⎩⎨⎧p=t−mq−p=s−tr=t−2s
⎩⎨⎧p=t−mq−p=s−tr+q−p=−s
⎩⎨⎧p=t−mr+2q−2p=−tr+q−p=−s
⎩⎨⎧m=t−pt=2p−2q−rs=p−q−r
⎩⎨⎧m=p−2q−rt=2p−2q−rs=p−q−r
Taking into account that f:R3→R is a linear function, we conclude that
Finding a professional expert in "partial differential equations" in the advanced level is difficult.
You can find this expert in "Assignmentexpert.com" with confidence.
Exceptional experts! I appreciate your help. God bless you!
Comments