There are numerous essays and newspaper blurbs lamenting the poor state of mathematical education in the US. Here is a typical example: Presidential panel bemoans state of math education.
What I see as the problem is that advanced mathematics is introduced in language that is unfit to inspire any but the few that were genetically destined to be mathematicians (or physicists).
Ask a recent college grad what an Eigen value or Eigen vector is. I give you 100:1 odds you'll get a blank stare. Okay now ask them to read this explanation from a popular Math web site. I bet their face will be even blanker. Now ask them to read this wonderful little explanation. Chances are the lights came on.
This is not to say that the later explanation will allow a person to do the math. But this is certainly where Math education, even at the highest levels, should begin. Illustrate why the problem is important, give a sensory picture to go along with the abstractions. Some might believe that this is how most Mathematicians teach but that is simply not the case. Mathematics is a very macho profession and many mathematicians believe its beneath them to offer intuition prior to rigor. The sad truth is many of them could not come up with compelling intuitive explanations even if they wanted to. It was not the way they were taught either.
Showing posts with label Linear Algebra. Show all posts
Showing posts with label Linear Algebra. Show all posts
Sunday, March 16, 2008
Saturday, February 9, 2008
Ubiquitous Eigenvectors and Quantum Computing
Recently I have been studying Quantum Computation (QC). If you want to get anywhere with QC you need to master Linear Algebra and Vector Spaces since QC is baically an exercise in applied vector space theory.
Being a bit rusty in the topic myself, I decided to pick up the book Finite-Dimensional Vector Spaces by Paul Richard Halmos. Although Halmos is one of my favorite math authors, I picked this book primarily because it has a Kindle edition. It turns out that another one of Halmos's books, Linear Algebra Problem Book, is a far better choice for the non-mathematcian. The later book walks you step by step through bite sized problems and provides hints (and also all the answers if you get stuck). A free resource with answers can also be found here.
One of the central mathematical techniques at the heart of Linear Algebra is the concept of Eigenvectors and Eigenvalues. The term Eigen is derived from German and means "characteristic". An Eigen Decomposition is a method of reducing a square matrix to into a constant (eigenvalue) and a vector (eigenvector). This decompostion is central to many problems in physics.
It turns out that the study of Eigen decompostion can yield deep insight into problems that are in the realm of computer science. Consider, for instance, this paper about Google's page rank algorithm and the Eigenface technique for facial recognition.
Coming to grips with the mathematics behind Vector Spaces is one of the single most rewarding experiences for anyone interested in advanced problems in computer science. It is a must if you ever want to graduate from the comprehension of clasical algorithms to the comprehension of quantum algorithms. However, if are curious about QC but the thought of learning advanced linear algebra sounds like too big of a comitment, then you might want to check out Quantum Computation explained to my Mother. This is the most approachable paper I have ever read on the topic that is also mathematically accurate.
Being a bit rusty in the topic myself, I decided to pick up the book Finite-Dimensional Vector Spaces by Paul Richard Halmos. Although Halmos is one of my favorite math authors, I picked this book primarily because it has a Kindle edition. It turns out that another one of Halmos's books, Linear Algebra Problem Book, is a far better choice for the non-mathematcian. The later book walks you step by step through bite sized problems and provides hints (and also all the answers if you get stuck). A free resource with answers can also be found here.
One of the central mathematical techniques at the heart of Linear Algebra is the concept of Eigenvectors and Eigenvalues. The term Eigen is derived from German and means "characteristic". An Eigen Decomposition is a method of reducing a square matrix to into a constant (eigenvalue) and a vector (eigenvector). This decompostion is central to many problems in physics.
It turns out that the study of Eigen decompostion can yield deep insight into problems that are in the realm of computer science. Consider, for instance, this paper about Google's page rank algorithm and the Eigenface technique for facial recognition.
Coming to grips with the mathematics behind Vector Spaces is one of the single most rewarding experiences for anyone interested in advanced problems in computer science. It is a must if you ever want to graduate from the comprehension of clasical algorithms to the comprehension of quantum algorithms. However, if are curious about QC but the thought of learning advanced linear algebra sounds like too big of a comitment, then you might want to check out Quantum Computation explained to my Mother. This is the most approachable paper I have ever read on the topic that is also mathematically accurate.
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