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Page 2

Sunday January 2, 2011 6:25 am Lethbridge

This page last updated on: Monday, January 10, 2011 7:59 PM

B. Actual Learning Activities

7:00 am

I will begin today by spending some time with Linear Algebra.

Linear Algebra for Dummies (2009)

Mary Jane Sterling

The first step is to review what I was doing a couple of weeks ago. My notes from December 6 are immediately following.

December 6 notes:

I read chap. 3 Mastering Matrices and Matrix Algebra yesterday. Today is a time for consolidation. Mathematics is largely about memory. One needs to have the various concepts at one's fingertips so the probability of a new connection or relationship is increased. Memory is largely about concepts and terminology. Facility is about using these concepts to accomplish some goal. Understanding is about connections.

Chap. 2 The Value of Involving Vectors

Here are the key terms:

• vector
• size of a vector
• scalar multiplication of a vector
• geometric representation of scalar multiplication
• vector addition
• magnitude of a vector
• Cauchy-Schwarz inequality
• inner product (dot product)
• test for orthogonality of two vectors
• computing the angle between two vectors

Chap. 3 Mastering Matrices and Matrix Algebra

Here are the key terms:

• matrix
• size (dimension) of a matrix
• adding & subtracting matrices
• scalar multiplication of a matrix
• multiplying two matrices
• additive identity matrix
• multiplicative identity matrix
• square matrix
• triangular matrix
• diagonal matrix
• singular & non-singular matrices
• matrix addition is commutative & associative
• matrix multiplication is not commutative
• matrix multiplication is associative
• matrix multiplication is distributive
• transposing a matrix
• the transpose of the sum of 2 matrices
• the transpose of the product of 2 matrices
• computing the inverse of a 2x2 matrix
• computing the inverse using row reduction

This was a good review. I had forgotten much of the detail but it quickly came back with the descriptions and examples in this chapter. Not only do I now know how to compute the dot product, but I know why I might want to do so.

Now for today's notes. What did I forget? Cauchy-Schwarz inequality inner product (dot product) test for orthogonality of two vectors computing the angle between two vectors computing the inverse of a 2x2 matrix. The first four items involve vectors, the last item involves the special case for 2x2 matrices. My bookmark indicates that I have completed reading chapters 4 & 5 as well, but I have yet to make any notes for these. Now to begin.

The Cauchy-Schwarz inequality [p. 32] is about the distances involving two vectors. Essentially it says in vector notation that the length of the hypotenuse of a triangle is less than or equal to the sum of the lengths of the other two sides. The notation involves using a pair of vertical lines on each side of the name of the vector to represent the length of the vector.

The inner product (dot product) of 2 vectors is a scalar. If the value is zero, then the two vectors are orthogonal to one another. If the inner product is not zero, then one can use another simple formula involving the value of the inner product to determine the angle between the two vectors. [p. 39]

The inverse of a 2x2 matrix can be computed using a relatively simple formula. (This is where handwriting is easier than computer-based textual systems.)

This is also where a computer language called APL is superior to other notational systems. I have been browsing the Web for information on APL on a Mac and found the following site: http://www.microapl.co.uk/apl/index.html

The inverse of a larger (square) matrix can be computed using elementary row operations on an augmented matrix which has an Identity matrix attached to the right side of the original matrix. Use elementary row operations to change the original matrix to an Identity matrix and the result to the right side augmented matrix will be the inverse matrix.

Moving on ... :

Chap. 4 Getting Systematic with Systems of Equations

Here are the key terms:

• system of (linear) equations
  • one solution
  • infinite solutions (use a parameter)
  • no solution
• 2 equations in 2 unknowns (2 lines)
• 3 equations in 3 unknowns (3 planes)
• inconsistent systems (i.e. no solution)
• solving a system of equations using algebra
  • 2 equations
  • 3 equations
  • n equations
• solving a system of equations using matrices
  • 2 equations
  • 3 equations
  • n equations
• coefficient matrix
• constant matrix
• inverse of coefficient matrix

This chapter is about the actual manual steps for solving systems of equations. However much of what is interesting about linear algebra is about underlying structures and relationships. We are interested in general statements that apply to all (or of a particular subset) vectors or matrices in 2, 3 or n dimensions. And when we move beyond 3 dimensions the patterns of symmetry become fascinating. And we begin to discuss groups, ...

Part II Relating Vectors and Linear Transormations

Chap. 5 Lining Up Linear Combinations

Here are the key terms:

• linear combinations of vectors
  • "Given a set of vectors with the same dimensions, many different linear combinations may be formed. And, given a vector, you can determine if it was formed from a linear combination of a particular set of vectors." [p. 88]
• reduced row echelon form
• vector set
• the span of a set of vectors
  • the set of all possible linear combinations
• "a span may be all encompassing, or it may be very restrictive." [p. 98]
• "A common question in applications of linear algebra is whether a particular set of vectors spans R*2 or R*3." [p. 101]

The concept of a span is relatively simple. The basic idea involves working with linear combinations of vectors. The next chapter examines the simple matrix equation Ax = b, where x and b are vectors and A is a matrix. The underlying idea is to see what types of general statements are possible.

tags: mathematics, vectors, matrices, linear algebra

9:20 am

The last two hours have just flown by. A great way to begin the morning. And I am intrigued with the possibility of getting a version of APL working on my Mac.

6:00 PM

Model Trains: I have spent 2 hours running Ship It! and Deluxe Balancer software. I have eliminated all of the diagnostic problems and reduced the number of warnings to 7. In addition I think I have both my passenger train and my unit coal train scheduled properly so the consist remains stable.

8:30 PM

I have finished reading "The Sentimentalists". I liked it, on two levels. One was the level and style of writing. The other was the implicit message that often there is no truth underlying events. Not a tidy ending, but a real one.

Summary

Description	Done
Mathematics: chaps. 4 & 5 of "Linear Algebra for Dummies"	Yes
Work on settings for Ship It! software.	Yes
Continue reading "The Sentimentalists" by Johanna Skibsrud	Yes

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