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Linear ALgebra questions

  1. May 20, 2012 #1
    I am trying to write a syllabus for a linear algebra course and I wanted to start with a set of questions that a first linear algebra course should address. So far I have the following which I think should be answered regarding linear systems:
    • Are there any solutions?
    • Does the system have no, unique or an infinite number of solutions?
    • How can we find all the solutions if they exist?
    • Is there some sort of structure to the solutions?
    Are there any questions we tend to answer on a linear algebra course?
     
    Last edited: May 20, 2012
  2. jcsd
  3. May 30, 2012 #2
    Maybe this is already covered by one of your questions, but are you going to get into vector spaces at all? My intro LA class made us do vector spaces and then abstract vector spaces as well.
     
  4. May 30, 2012 #3

    chiro

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    Hey matqkks.

    You might want to discuss what linear things are and how they are used in various areas of mathematics. As 20Tauri pointed out, vector spaces are one natural way of capturing linearity in an abstract way.

    In terms of understanding linearity, you can understand this by considering the Euclidean co-ordinate system and how this relates to 'linear combinations' of independent things. Because of this euclidean connection, the linear algebra has a natural connection with geometry and applications for this relate to say orthogonality which is used in least squares which is applied to many areas including regression for statistics. Other geometric connections are with dot products and cross products as well as the foundations for quantum mechanics.

    Basically the linear framework is such that we can take something and write it as a linear combination of basis vectors. Then you can explain that a particular segment of linear algebra (spanning, basis sets, row-reduction, REF, finding solutions, rank, rank-nullity) is concerned with doing this exact thing by reducing systems of linear equations down to the simplest possible description.

    Also differential operators are linear operators and have a natural interpretation in vector and multivariable calculus. This can be briefly mentioned so that people understand the matrix stuff they will see in a Calculus III course.
     
  5. May 30, 2012 #4
    Here's what we did in my first course. It was lower division.

    Systems of Linear equations and solving them

    Linear Transformations

    Determinants

    Vector Spaces

    Eigenvalue Problems

    Inner Products
     
  6. May 31, 2012 #5
    My last sentence should read:
    Are there any other questions we tend to answer on a linear algebra course?
     
  7. May 31, 2012 #6

    chiro

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    Maybe show that a way of decomposing something into linearly independent things and that you can always do this systematically for a fixed-dimension 'thing'.

    This is important because decomposition is what analysis is all about: i.e. the 'breaking down' of things.

    When people understand this, they will see that you can take something and 'break it apart' in many ways (different bases) and then use rank-nullity, dimension, spanning and so on to tie this concept all together.

    When you show the students how to 'break-down' general things, they will get math as opposed to just showing them lots of formulas and have them look at you with a blank face.
     
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