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Page 2

Friday January 14, 2011 5:30 am Lethbridge

This page last updated on: Saturday, January 15, 2011 5:48 AM

B. Actual Learning Activities

8:30 am

I have finished viewing Lecture 1 of the MIT course 18.06 Linear Algebra with Gilbert Strang, the author of the textbook I purchased recently. I loved it. His perspective of column vectors is new to me, but I can see the power of it. At the moment I am not sure of how well these YouTube videos will integrate with the textbook, but we shall see.

If I were actually attending these lectures, I would be taking notes. So why not now? A partial answer might be because I can re-view the video at any time. But the real value of making a few notes is to try to capture the important points. I must view this again and see what notes I can make while it is unfolding.

10:40 am

Introduction to Linear Algebra (2009)

Gilbert Strang

I have watched Lecture 1 of the MIT course 18.06 Linear Algebra with Gilbert Strang. First, I simply watched it. Then I watched it a second time, making some scribbled notes. Now I am going to try to recompose them into a coherent set of web notes. This lecture appears to cover some of the material that is in chapter 1 of his textbook.

Lecture 1 MIT course 18.06 Linear Algebra

• The fundamental problem of linear algebra is to solve (and to understand) systems of linear equations in n unknowns.
  
  
• Three ways of considering such a system are:
  • by means of a row picture (i.e. by considering one row [equation] at a time)
  • by means of a column picture (i.e. by considering one column [vector] at a time)
  • by means of a matrix equation.
    
    
• Row picture (2 dimensions) (2x -y = 0)
  • look at 2 points that satisfy the first equation (try x = 0, then y = 0)
  • then consider ALL of the points that satisfy this equation (a straight line through the two points)
  • now consider ALL the points that satisfy the second equation (another straight line)
  • if the two lines intersect then the coordinates of the point are the solution that satisfies both equations.
    
    
• Column picture (2 dimensions)
  • form a vector equation by rewriting the two equations in vector form
  • the question then becomes "what is the right linear combination of the two vectors to produce the third vector?"
  • what is the corresponding picture for this situation? Draw the first vector then draw the second vector from the end point of the first vector. What are the coordinates of the final point?
    
    
• Consider ALL of the combinations of the first vector and ALL of the combinations of the second vector. This usually will be ALL of the points in the xy plane.
  
  
• Begin by trying to understand the equations, then try to solve them.
  
  
• Consider 3 equations in 3 unknowns
  • (3x3 matrix) times vector = vector
  • Row picture: the 3x3 matrix describes 3 planes (1 for each row in the matrix). This is already starting to be difficult to visualize.
  • Column picture: a linear combination of 3 vectors. This is easy to visualize. It is just a linear combination of 3 vectors, each with 3 elements. From there it is easy to extend this to 9 dimensions or n dimensions.
  • Thus the column picture is much more powerful than the row picture!
    
    
• Consider 2 equations in 2 unknowns. "It is the right place to start."
  
  
• "What combination should I take? Why not take the right combination?"
  
  

  How does one understand a linear system of equations? Row pictures (one equation at a time) Column pictures (vector equations) Matrix notation Putting it in your own words How about a semantic (concept) chart? Work through some examples Then keep iterating through all of the above, enriching the connections and relationships.

  
  
  
• Do certain linear combintions fill a space?
  
  
• When could it go wrong? Under what conditions will a situation fail? (e.g. when all vectors are in the same plane.)
  
  
• Imagine that we have ...whatever. Too difficult? Then pretend you can imagine it.
  
  
• "Generate a random 9x9 matrix. It is almost certainly non-singular."

Of course. None of the 9 column vectors would likely be a linear combination of the others. Neat.

• How do you multiply a matrix by a vector?
  • Row view: commonly called the dot product
  • Column view: rewrite as a linear combination of column vectors and then evaluate.

It had never occured to me that there were two ways to do this! This alone made the lesson worthwhile.

 

Time 11:40 am Making these notes has been very worthwhile. I now want to try doing matrix vector multiplication both ways using APL.

 

5:00 PM

Much of the afternoon was spent with my Model Trains. It is so much more fun when everything runs smoothly. I really enjoy the switching activity. And the locomotives are running without any stalls even at very slow speeds. Great. Details at the Dales Depot website.

Summary

Description	Done
Technology: Lesson 5 of "Adobe Dreamweaver CS5 Classroom in a Book"
Mathematics: Lecture 1 of MIT course 18.06 Linear Algebra by Gilbert Strang	Yes
Model Trains: Continue operational runs for Session 1	Yes
Continue reading "Freedom" by Jonathan Franzen

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