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## Homework Statement

I am part way done with this problem.... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now.

W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it.

a) Find the basis of {v1, v2} for this subspace.

b) These basis vectors are basis vectors for what eigenvalue?

c) Show that n=<1,1,1> is orthogonal to {v1 ,v2}

d) Show that n is orthogonal to all of W

e) n is an eigenvector for T for what eigenvalue

f) Using matrix with eigenvectors and one for eigenvalues, find the standard matrix of T.

## Homework Equations

## The Attempt at a Solution

I am part way done with this problem.... I don't know how to solve part e or part f.

W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it.

a) Find the basis of {v1, v2} for this subspace.

I found the basis of W to be v1=<-1,1,0> and v2=<-1,0,1>

b) These basis vectors are basis vectors for what eigenvalue?

I found the eigenvalue to be zero after multiplying W with v1 and v2.

c) Show that n=<1,1,1> is orthogonal to {v1 ,v2}

I just took the dot product of n*v1 and n*v2 and both were zero

d) Show that n is orthogonal to all of W

My reasoning is that n is orthogonal to all of W because W is a linear combination of {v1 ,v2}

e) n is an eigenvector for T for what eigenvalue

??????

f) Using matrix with eigenvectors and one for eigenvalues, find the standard matrix of T.

??????