Please see the attached file for the fully formatted problems.

Problem
a. quote a theorem which guarantees that there exists an orthogonal basis for (with standard inner product) made up of eigenvectors of matrix

b. Find such a basis .
c. Represent the quadratic form by a symmetric matrix. Is Q positive definite? Justify your answer.

My explanation
a. I am not sure this is right but here is theorem that we learned in class.
Theorem 1
Let T be a linear operator on a finite-dimensional real inner product space V. Then T is self-adjoint if and only inf there exists an orthonormal basis for V consisting of eigenvectors of T.
( we know that A is self-adjoint if )
Since A is self-adjoint I guess I can use theorem 1???
What is the connection between T and A??

b. To find a basis made up of eigenvectors of matrix A, first of all, we know that we need to find the eigenvalues. Using det(tI-A) I found that and . (please check that it is correct).
For , I found eigenvectors (-1,1,0) and (-1,0,1) ,and for , eigenvector is (1,1,1) . (please check that it is correct).
here is my question, I know that (-1,1,0) and (1,1,1) are orthogonal and so (-1,0,1) and (1,1,1). Also another theorem tells that if and are distinct eigenvalues of T with corresponding eigenvectors x and y , then x and y are orthogonal. but how about (-1,1,0) and (1,1,1)??? Obviously they are not orthogonal. Is it because of that they are eigenvectors of same eigenvalues??

c. here is the definition of the positive definite.

A linear operator T on a finite-dimensional inner product is called positive definite if T is self-adjoint and <T(x),x> >0 for all .
An n x n matrix A with entries from R or C is called positive definite if is positive definite.

Please represent the quadratic form by a symmetric matrix. And verify that Q positive definite.
Thank you.

Orthogonal basis is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

1. Determine the eigenvalues, determinant, and singular values of a Householder reflector. Give algebraic proofs for your conclusions.
2. Suppose Q E C^n, llqll2 = 1
Set P = I - qq^H.
(a) Find R(P)
(b) Firrd l/(P).
(c) Find the eigenvalues of P.
Prove your clairns.
3. Let A E C^(m*n)}, m (greater or equal to) n, with ran

1) For which values of k are the following vectors u and v orthogonal?
a) u = (2,1,3) , v = (1,7,k)
b) u = (k,k,1) , v = (k,5,6)
2) Let u,v be orthogonal unit vectors. Prove that d(u,v) = 2^(1/2)
(The questions are unrelated)

R stands for the field of real numbers. C stands for the field of complex numbers.
1. Let T be a linear transformation from the set P2(R) of all polynomials of degree at most 2 into itself.
T: P2(R) --> P2(R), given by T(f) = f' - f'', fEP2(R),
where f' is the first and f'' is the second derivative of f.
(a) Find the null

In C[-pi, pi] with inner product 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 LinearAlgebra With Applications by Steven J. Leon, Sixth Edition. Than

1) Find the closest point to y in the subspace W spanned by v1 and v2.
y=
v1=
v2=
#2) Let y, v1, and v2 be as in exercise #1. Find the distance from y to the subspace of R4 spanned by v1 and v2.
#3) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal bas

Please show work.
Consider R^p with the standard inner product....Prove that the orthogonal projection...is a linear transformation...Is the transformation one-to-one (injective)? Is the transformation onto (surjective)? Justify your answers.

Let T be a linear map on R^2 defined by T(x,y) = (4x - 2y, 2x + y).
Calculate the matrix of T relative to the basis {α1, α2} where α1 = (1,1) , α2 = (-1,0).

Please show the complete steps.
1. Let T: P2 -> P1 be the linear transformation defined by T(p(x)) = p'(x) + p(x). Show that T is linear.
2. Find the matrix for the transformation T given in problem 1 with respect to the standard basis {1, x, x^2}. Then, find all eigenvalues and corresponding eigenvectors for T.
3. True