MATH 115 University of California LA Gram Schmidt Algorithm Questions

I’m working on a linear algebra multi-part question and need an explanation to help me understand better.

1. As we have defined the Gram-Schmidt algorithm, we assume that the input is a list of independent vectors. What happens if you do the Gram-Schmidt algorithm to a list of dependent vectors? Be specific in identifying where the algorithm behaves differently, and what the result of the algorithm is in this case.

2. For each of the following statements, i) Find the negation of the statement, and ii) determine if the original statement is true or false. Justify your answer.

(a) If dim(V ) > 1 and T ? L(V ) has exactly one eigenvalue, then T is not diagonalizable.

(b) Every linear operator over a complex vector space is diagonalizable.

3. Suppose that if v, w are vectors in V , and T ? L(V ) is an operator such that

T (v) = v ? w and T (w) = v + w

(a) Show that span(v, w) = span(v ? w, v + w)

(b) Now assume that v,w are a basis for V. Explain why part a, shows that T is an

isomorphism.

4. Consider the following statement:

If {v1,…,vn} are independent vectors in some vector space V and v is not in the span of v1,…,vn, then {v1,…vn,v} must also be independent.

(a) Find the contrapositive of this statement.

(b) Prove this statement.

(c) Use this statement to prove, by induction, that for any n ? 1, the functions 1, cos(x), cos(2x), . . . , cos(nx)

are independent vectors in the vector space of continuous functions.

5. Consider the operator T ? L(R^n) which maps (x1, . . . xn) ??? (xn, x1, x2 . . . xn?1). Observe that T^n = I, where I is the identity transformation. Use this fact to answer the following.

(a) Show that 1 is always an eigenvalue of T . What is E(1, T )?

(b) Show that -1 is an eigenvalue of T iff n is even. What is E(?1,T)?

(c) Show that there are no other real eigenvalues of T (besides 1 and ?1).

(d) For what n is T diagonalizable? You must justify both why T is diagonalizable when n satisfies some condition, and why T is not diagonalizable (over the reals) when n does not satisfy this condition.

(e) Describe the adjoint T ^*. Is T a normal operator?

6. Consider M^2×2(R), the vector space of 2×2 matrices with real entries, and the subspace U consisting of all symmetic matrices.

(a) Show that U is 3 dimensional by establishing an isomorphism between U and a 3 dimen- sional space.

(b) Consider the inner product on M^2×2(R) defined by ?A,B? = tr(A^TB). Find a matrix X such that X is orthogonal to the identity matrix under this inner product.

(c) Explain how you know that span(X) = U^? in M^2×2(R)