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

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

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

Show that the functions x and x^2 are orthogonal in P5 with inner product defined by (

=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.

B1) This question concerns the following two subsets of :
(a) Show that , and find a vector in that does not belong to T. [3]
(b) Show that T is a subspace of . [4]
(c) Show that S is a basis for T, and write down the dimension of T. [7]
(d) Find an orthogonalbasis for T that contains the vector .

Given a vector w, the inner product 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 inner product 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 inner product. Compute the values of

(See attached file for full problem description)
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For the A-matrix:
5x1 + 9x2 + 2x3 = 24
9x1 + 4x2 + x3 = 25
2x1 + x2 + x3 = 11
construct an orthonormal basis with a1 and then a2 and then a3. Next, expand the given vector b in terms of those vectors.
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See the attached file.
Let Beta = {x_1, ..., x_n} be a basis for a vector space V, and let P be the mapping P((a_1)(x_1) + ... + (a_n)(x_n)) = (a_1)(x_1) + ... + (a_k)(x_k).
a) Show that Ker(P) = Span({x_k+1, ..., x_n}) and Im(P) = Span ({x_1, ..., x_k})
b) Show that P^2 = P
c) Show conversely that if P:V --> V is an