Announcements and Homework Hints
Homework #1 hints.
- Problem #1.
You only need to answer yes or no, unless asked to prove something. I assigned this problem just to get you to review what a basis was, what the dimension of a space meant, and how many linearly independent vectors can be in a basis.
- (a). You can safely assume that the dimension of C^n is is n, and that each v_n is in C^n.
- (c). The fact that span{v_1, ..., v_n}=C^n, means that any vector in C^n can be written as a linear combination of v_1, v_2, ..., v_n.
- (e). Just give an answer, no need to justify or explain. Note that the v_1 and v_2 are the same v_1 and v_2 as the rest of the problem.
- (f). Assume that there are two sets of coefficients such that c_1v_1 + ...+ c_n v_n = b. Show that they cannot be distinct if the v`s are linearly independent, using the definition of linear independence given in (a).
- 1.1: more precisely, at each of the seven steps, put a new matrix on either side of the product of matrices from the previous step. The point of this problem, is just to exercise the brain a little, regarding how operations on matrices can be accomplished by matrix multiplication.
- 1.4 Note how the eight equations are a matrix vector equation. You do not need to know anything about the functions themselves, or find any inverses in this problem.
- 2.3
- (a). Using what you know about adjoints and inner products, and hermitian matrices (A^* = A), try and show that the lambda^* = lambda, (for any eigenvalue lambda of A). That is, the complex conjugate of lambda equals lambda itself, which implies that lambda must be real.
- (b). As above, use what you know about adjoints, hermitians, and inner products. For two different eigenvectors, x and y, (with two different eigenvalues lambda_1 and lambda_2) try and show that the inner product (x,y)=0. Very good hint: note that the equation (lambda_1 - lambda_2)(x,y)=0, proves that either x and y are orthogonal or lambda_1 = lambda_2. Try and establish that equation.
- 3.3 For the matrix inequalities, replace the matrix norm by its definition as a supremum (max) of a vector norm before trying to establish the inequality. Then you can make use of (a) and (b) in your inequalities, since ||Ax|| is just a vector norm. Also, don`t let the sup bother you. If g(x)>= f(x) for all x, then sup g(x) >= sup f(x). For finding examples of matrices that satisfy equality, read the examples in the text that point out that the infinity norm of a matrix is the maximum of row sums. Good Hint: then look for matrices that are all zeros except for ones in a single column or row.