Department of Electrical and Computer Engineering

The ECE 556 FAQ

Answers to Frequently Asked Questions

The official prerequisite for ECE 556 is Math 417: Introduction to Abstract Algebra, or equivalent, or consent of instructor.

Unofficially, you are expected to have a good grasp of linear algebra, at a level more akin to what is taught in MATH 418 Introduction to Linear Algebra than in MATH 415 Linear Transformations and Matrices. For those unfamiliar with these courses and their different emphases, here is a simple explanation. If you cannot think of vectors except in terms of n-tuples, and linear transformations except in terms of matrices, then you should expect to have considerable difficulty in ECE 556. As a self-test, try the exercises below and see if you can write down the proofs without using n-tuples and matrices.

  • Suppose that x and y are vectors in a vector space V, and that M is a subspace of V. Let H denote the subspace spanned by M and x; and K the subspace spanned by M and y. Prove that if y is a member of H but not of M, then x is a member of K.

  • Let V denote an n-dimensional vector space. Prove that the set of all linear transformations from V to V constitutes a vector space W. What is the dimension of W? What is 0, the zero element of W?

    Let A be a linear transformation in W. Prove that the set of all linear transformations B in W such that AB = 0 is a subspace of W.

  • Your engineering professor states that a transformation is called linear if the transform of a sum is the sum of the transforms, that is, the transformation L on a vector space V is called a linear transformation if

    L(x + y) = L(x) + L(y) for all vectors x and y in V.

    Do you agree? If not, state clearly what you understand to be the definition of a linear transformation, and give an example of a transformation L that satisfies the above, but is not a linear transformation according to your understanding of the phrase.

Back to the list of questions or to the ECE 556 Home Page

  • So, what about MATH 417?
Why don't the ECE 556 instructors demand that all students have taken the official prerequisite Math 417 or equivalent? The reason is that Math 417 is not strictly necessary for understanding the ECE 556 material. Typically, introductory courses in abstract algebra such as Math 417 cover groups, rings, domains, fields and field extensions. Having a good understanding of these concepts will help you achieve a solid understanding of the ECE 556 material as well. But, such a detailed understanding of abstract algebra is not strictly necessary. It is also possible to learn coding theory (as many students have done in the past) with a strong background of linear algebra together with the material on finite fields that is presented in ECE 556. Note that Math 417 and the like are not directly helpful in that finite fields are discussed only briefly, if at all, in typical undergraduate courses, usually as a simple or trivial example that illustrates the general theory. On the other hand, ECE 556 requires you to understand more than this about finite fields: enough of the theory at a level where you can understand and carry out by hand the various arithmetic operations in finite fields, and understand (and even design) simple logic circuits (or write short program fragments) to implement such arithmetic operations as parts of encoders and decoders for various codes. The necessary background for these skills is taught in the course itself.

Back to the list of questions or to the ECE 556 Home Page

  • Is there life after ECE 556?

Surprisingly enough, yes. Many students not only survive ECE 556, but some actually use what they have learned in ECE 556 in later life, while others go on to successful careers in industry and academia. More specifically, ECE 556 is a prerequisite for certain offerings (Codes for Communications and Cryptography, and Euclidean-Space Codes and Constellations) of ECE 559: Topics in Communications, and ECE 560 VLSI in Signal Processing and Communications. It is also helpful (though not strictly necessary) as general background for some other offerings of ECE 559: ( Codes on Graphs, and Communications III), as well as for ECE 563/STAT 563/CS 578: Information Theory.

Back to the list of questions or to the ECE 556 Home Page