PHY 6645 Fall 2002 – Homework 1
Due by 5 p.m. on Friday, September 6. Partial credit will be available for
solutions submitted by 5 p.m. on Monday, September 9.
Answer all questions. To gain maximum credit you should explain your reasoning and show
all working. Please write neatly and include your name on the front page of your answers.
1. Determine whether each of the following examples constitutes a linear vector space
when addition and scalar multiplication are defined in the standard fashion and scalar
multiplication is taken over the field of real numbers. For each example that is a vector
space, determine the dimension of that space.
(a)
(b)
(c)
(d)
(e)
The set of all integers.
The set of all real numbers.
The set of traceless 2 × 2 real matrices.
The set of orthogonal 2 × 2 real matrices.
The set of all polynomials of x (x real) of fourth order or less, with complex
coefficients.
(f) The set of all infinite vectors (x1 , x2 , x3 , . . .), where (i) xn is real for all n, and (ii)
there is a finite number of nonzero components, i.e., there exists a finite N such
that xn = 0 for all n ≥ N .
2. Shankar Exercise 1.3.4.
3. Shankar Exercise 1.6.2.
4. Shankar Exercise 1.6.6.
5. Shankar Exercise 1.8.3.
6. Shankar Exercise 1.8.4.