Show that the functions x and x^2 are orthogonal in P5 with inner product defined by ( <p,q>=sum from i=1 to n of p(xi)*q*(xi) ) where xi=(i-3)/2 for i=1,...,5.

Show that ||X||1=sum i=1 to n of the absolute value of Xi.

Show that ||x||infinity= max (1<=i<=n) of the absolute value of Xi.

Thank you for your explanation.

Solution Summary

Vector relations are proven given an inner product. The solution is detailed and well presented.

Can you help me with the following:
Let ubar and vbar be nonzero vectors in 2 or 3 space and let k=||u|| and m=||v||. Show that the vector Wbar=mubar +kvbar bisects the angle between ubar and vbar.
Please show all steps in detail for my reference.

In C[-pi, pi] with innerproduct defined by (6), show that cos mx and sin nx are orthogonal and that both are unit vectors. Determine the distance between the two vectors.
(6) (f,g) = (1/pi)* the integral from -pi to +pi of f(x)g(x)dx
This is all from Linear Algebra With Applications by Steven J. Leon, Sixth Edition. Than

Given a vector w, the innerproduct of R^n is defined by:
=Summation from i=1 to n (xi,yi,wi)
[a] Using this equation with weight vector w=(1/4,1/2,1/4)^t to define an innerproduct for R^3 and let x=(1,1,1)^T and y=(-5,1,3)^T
Show that x and y are orthogonal with respect to this innerproduct. Compute the values of

Could you clarify what constitutes a spanning set and a basis? Also how does one test to see if a set of vectors is a spanning set and if it is a basis?

I have two questions that I need help with.
1) How would you find a basis of the kernel, a basis of the image and determine the dimension of each for this matrix? The
matrix is in the attachment.
2) Are the following 3 vectorslinearly dependent? (see attachment for the three vectors) How can you decide?
I hope y

Question 4
Find the work done by the force
F (x,y,z) = -x^2y^3 i + 4j + xk
on moving charged electric particle along the path given by the equation
r (t) = 2cos t i + 2sintj + 4k,
where the parameter t varies from pi/4 to 7pi/4.
Question 5
Displacement of the spring system with friction is described by the differenti

Please help me solve the following linear algebra questions involving linear transformation and matrices. (see attached)
? Let and let . Define a map by sending a vector to .
a) let and be the standard basis vectors of V. let , and be the standard basis vectors of W. Find the matrix of T with respect to