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Linear transformations, Matrix diagonalization and Orthogonal projections

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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 space N(T) of T. What is the dimension of N(T).
(b) Find the matrix representation [T]e of T in the basis below:ch

2. Let a = [2 0; 1 -3]
(a) Find the characteristic polynomial f(t) = det (A - tI).
(b) Find the eigenvalues and corresponding eigen vectors of A.
(c) Is A diagonalizable? Justify your answer by citing a proper theorem.
(d) Find an invertible matrix Q such that D = Q-1AQ is diagonal.

3. Let w1 = [-1; 1; 0] and w2 = [0; 1; 2], w1, w2 E R^3.
(a) Use the Gram-Schimdt process to orthogonalize, and then to orthonormalize, the vectors w1, w2.
(b) Let W = span{w1, w2}. Find the orthogonal projection of y = [0; 1; 0] to W.

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Solution Summary

The solution explains how to find the matrix representation of a linear transformation, how to find eigenvalues and eigenvectors, diagonalizing a matrix, Finding orthogonal basis and how to find orthogonal projection onto a vector space.

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Eigenvalues and Transformations

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 or False: For a given transformation T, if T(0) = 0, then T is linear. If you answer true, prove the claim. If you answer false, find a counter example and show the transformation is not linear.

4. Let f = f(x) and g = g(x) be two functions in C[a,b] and define <f,g> = Integral with limits a to b of f(x)g(x) dx. Show <f,g> defines an inner product on C[a,b].

5. Assume the Euclidean inner product on R^3. Apply the Gram-Schmidt process to transform the basis vectors u1 = (1,0,1), u2 = (0,1,1), and u3 = (0,0,1) into an orthogonal basis {v1, v2, v3}.

6. Let T: R^3 -> R^3 be the orthogonal projection on the yz-plane. Find the matrix for T relative to the standard basis for R^3, and then find the matrix for T relative to the basis v1 = (1,0,0), v2 = (1,1,0), and v3 = (1,1,1).

7. Let
A = | 3 2 |
| 2 0 |.

By inspection, what property of A indicates that there exists a basis consisting of two orthonormal eigenvectors of A? Find an orthogonal matrix P that diagonalizes A and show that the diagonal matrix has eignevalues of A along its diagonal. (Hint: Make sure your eignevectors are orthonormal).

8. Find a basis B relative to which the transformation T: P2 -> P2 given by T(a+bx+cx^2) = a + 2cx + (3b+c)x^2 is diagonal. Then find [T]_B.

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