Math410 (Bayly) Homework 2 (due Monday 2 February)

Section 1.3: 1, 2, 3, 4, 5, (6). In (6) determine for which values of a there are 0, 1, or an infinite number of solutions (and in the last case express in terms of a free variable).

Section 1.5: 4, (5), 11, (19)

Section 1.6: 1, (6), (11)

You know of course that there are nonzero 2x2 matrices that do not have inverses, and they don’t always commute, and crazy stuff like that. However if you consider the class of 2x2 matrices of the form , you’ll get a nice surprise. First note that this matrix can be written as where is the identity and is a new matrix you’ve never seen before. However, you can check (don’t turn in) that , which is kind of like the definition of the complex number i.

What I would like you to check (and turn in!) is that , just as for complex numbers . Therefore any nonzero matrix in this class does have an inverse, and it turns out that all the rules of complex arithmetic hold for matrices of this form. The discovery of this matrix representation, in terms of real numbers only, of complex numbers helped convince skeptics that complex numbers were indeed real.

Section 2.2: 3, (6), 7, (8), (9), 10

In (6) and (8), when there are infinitely many solutions, find the solution that has smallest Euclidean norm.

Extra Credit/ Honors Credit:

(1) After the discovery of complex numbers, many mathematicians searched for systems of numbers with more than two components. This search ultimately led to general matrix algebra in the 1850’s and later, but in 1843William Rowan Hamilton had a spectacular insight that a 4-component system was possible. He had spent much of the 1830’s trying to find a 3-component system, hoping to find a natural way to do algebra with three-dimensional space. But he, like everyone else, was unsuccessful. With our current knowledge of matrices and complex numbers, it turns out to be easy. The only tricky part is that we’re going to mess with the notation a bit. We’re going to denote the identity matrix by U (for unit) instead of I, and we’re going to define . I want you first to check that , so that I, J, K can all be thought of as square roots of -1. I’d also like you to check that , which you hopefully recognize as the vector cross-product relations for the coordinate unit vectors . (So he did succeed in developing a useful 3d algebra after all!) Last, I’d like you to confirm that any nonzero quaternion, a matrix of the form aU+bI+cJ+dK with real numbers a, b, c, d, has an inverse.