# Orthogonal basis and matrix, Gram-Schmidt process

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).

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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