Linear Algebra

Q5. By completing the square, sketch the following quadratic curve:

2. Explain why f(x) = x + 1 is not a linear transformation from R → R.

3. Find the standard matrix of the linear transformation from R2 → R²: (a) Reflection across the x-axis. (b) Clockwise rotation by 7/2 radians.

Problem 1 If A = [1/2 1-/4] and we shift to A-7l, What are the cigenvalues and eigenvalues and

Problem 2. Give an example to show that the eigenvalues can be changed when a multiple of one row is subtracted from another. Why is a zero eigenvalue not changed by the steps of elimination?

Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v, w. (a) Give a basis for the nullspace and a basis for the column space. (b) Find a particular solution to Ax=v+w. Also, find all solutions to Ax=v+w.

Problem 5. Choose the second row of A so that A has cigenvalues 4 and 7.

Problem 8. Write the most general matrix that has eigenvectors

Problem 6. (a) If A² = I, what are the possible eigenvalues of A? (b) If this A is 2 by 2, and not I or -1, find its trace and determinant. (c) If the first row is (3,-1), what is the second row?

Problem 7. Suppose the eigenvector matrix S has STS-¹. Show that A = SAS-¹ is symmetric and has orthogonal eigenvectors.