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1) Find the coordinate vector of the polynomial

p(x)=1+x+x^2

relative to the following basis of P2:

p1=1+x, p2=1-x, p3=1+2x+3x^2

?????

I wasnt sure how to work this problem out:

Does it start out as?

b1=1,t,t^2

b2=t,1,t^2

b3= 1+t, 1-t, t-t^2

2) Let X be the linear span of the vectors

(1,1,1,1) (1,1,1,0) (1,1,0,0)

in R^4. Find the orthonormal basis for X?

It is:

[[u1]]^2

[[u2]]^2

[[u3]]^2

u1=1/2(1,1,1,1)

u2=1/6(1,1,1,0)

u3=1/4(1,1,0,0)

3) Let X be the linear span of the vectors

(1,2,1,2) (1,2,1,0) (1,1,0,0)

in R^4. Find the orthogonal projection of the vector (1,1,1,1) on th esubspace X?

It is solved like this:

c1=(v,u1)/(u1/u2)=(1+2+1+2)/(1+4+1+4)

c2=(v,u2)/(u2/u2)=(1+2+1+0)/(1+4+1+0)

c3=(v,u3)/(u3/u3)=(1+1+0+0)/(1+1+0+0)

there for x=proj(v,x) = c1u1+c2u2+c3u3