Let #P_2# is real polynomial space with the highest degree is 2. Specify the #x^2"-coordinate "# of the base #" "{x^2+x,x+1,x^2+1}" in "P_2# ?

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Answer 1 · 2017-04-09T09:10:52

#(1/2,-1/2,1/2)#

I think you are asking for the coordinates of #x^2# using the given base.

We find:

#1/2(x^2+x)-1/2(x+1)+1/2(x^2+1) = x^2#

So the coordinates are: #(1/2, -1/2, 1/2)#

Answer 2 · 2017-04-09T09:13:27

#x^2 = 1/2(x^2+x)-1/2(x+1)+1/2(x^2+1)#

We have that

#x^2=alpha(x^2+x)+beta(x+1)+gamma(x^2+1)#

so

#x^2=(alpha+gamma)x^2+(alpha+beta)x+(beta+gamma)#

and we need

#{(alpha+gamma=1),(alpha+beta=0),(beta+gamma=0):}#

solving we have

#alpha=1/2,beta=-1/2,gamma=1/2# and finally

#x^2 = 1/2(x^2+x)-1/2(x+1)+1/2(x^2+1)#