1. Let V be an inner product space. Prove the Pythagoreantheorem: if v and w E V are orthogonal then ||v+w||^2 =||v||^2 + ||w||^2. (Suggestion: start with ||v+w||^2 = (v+w,v+w).)
2. Let V=P2, the polynomials of degree less than/equal to 2 withcoefficients in R, and let (.,.): VxV -> R be the map
(p,q) = p(0)q(0) + p(1)q(1) + p(2)q(2) for any two polynomialsp(x),q(x) E V. So, for example,
(x+2, x^2-1) = (0+2)(0^2-1) + (1+2)(1^2 - 1) + (2+2)(2^2-1) =10.
a) Show that the function(.,.) defines an inner product onV.
b) Is the set A=(1,x,x^2) an orthogonal set? (with respect tothe inner product above.)
c) Is the set B=(1,x-1,3x^2-6x+1) an orthogonal set?
d) Calculate ||1||^2, and ||3x^2-6x+1||^2
e) Is the set B an orthonormal set?
f) Write f=x^2-6x+12 as a linear combination of the vectors inB.
1. Let V be an inner product space. Prove the Pythagoreantheorem: if v and w E V are orthogonal then ||v+w||^2 =||v||^2 + ||w||^2. (Suggestion: start with ||v+w||^2 = (v+w,v+w).)
2. Let V=P2, the polynomials of degree less than/equal to 2 withcoefficients in R, and let (.,.): VxV -> R be the map
(p,q) = p(0)q(0) + p(1)q(1) + p(2)q(2) for any two polynomialsp(x),q(x) E V. So, for example,
(x+2, x^2-1) = (0+2)(0^2-1) + (1+2)(1^2 - 1) + (2+2)(2^2-1) =10.
a) Show that the function(.,.) defines an inner product onV.
b) Is the set A=(1,x,x^2) an orthogonal set? (with respect tothe inner product above.)
c) Is the set B=(1,x-1,3x^2-6x+1) an orthogonal set?
d) Calculate ||1||^2, and ||3x^2-6x+1||^2
e) Is the set B an orthonormal set?
f) Write f=x^2-6x+12 as a linear combination of the vectors inB.
