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MTH 4100 Linear Algebra and Matrix Methods Class Notes

  • This is an unofficial thread.
  • Class notes from Linear Algebra with Matrix Methods, Fall 2008, will be posted here.
  • The instructor is Alfred Friedland; the class meets M/W 6-715pm.
  • All students are encouraged to review the notes and comment in the event of errors and omissions.
  • The notes are in .doc format with Microsoft Equation objects embedded.
  • If someone wants to translate them into LATEX and post them here that would be most cool.
Note attachments to this post:
  • Statement of Policies and Expectations
  • Textbook Description
  • First Homework Handout
 

Attachments

  • 4100 Policies and Expectations.pdf
    1.2 MB · Views: 44
  • 4100 Textbook Description.pdf
    477 KB · Views: 67
  • 4100 Homework Handout.pdf
    641.9 KB · Views: 52
Wed, August 27, 2008

notes attached.

[well, depending on what you have in the way of Microsoft equation, the matrix notation can look a little funky. I'll attend to that in a bit.]
 

Attachments

  • 087208.doc
    27.5 KB · Views: 87
Wed, Sep 4, 2008

notes attached.

The diagonal elipsis in the matrix notation may, depending on the version of MSFT Equation you have, read like some funky character, but you should get the idea.
 

Attachments

  • 090308.doc
    68.5 KB · Views: 47
Mon, Sep 8, 2008

notes attached.

  • Homogeneous System of Linear Equations defined.
  • Coefficient Matrix
  • Augmented Matrix
  • Gauss-Jordan Elimination
  • Homework assigned from the handout (see introductory post below).
 

Attachments

  • 090808.doc
    68.5 KB · Views: 24
Wed, Sep 10, 2008

notes attached.

  • Gauss-Jordan Elimination, continued.
  • Operations on Matrices
    • Matrix Equality
    • Addition & Subtraction
    • Scalar Multiplication
 

Attachments

  • 091008.doc
    76.5 KB · Views: 24
Mon, Sep 15, 2008

notes attached.

  • Matrix Multiplication
  • Writing a System of Linear Equations as a Matrix Equation
Homework handout with the exception of inverse problems should be completed by Monday, Sep. 22.
 

Attachments

  • 091508.doc
    84.5 KB · Views: 23
Wed, Sep 17, 2008

notes attached.

  • Matrix Valued Polynomial Functions
  • Matrix Inversion
The entire homework handout should be completed.
 

Attachments

  • 091708.doc
    81.5 KB · Views: 25
Monday, Sep 22, 2008

notes attached.

  • Matrix Inversion, continued
  • Properties of Matrix Operations
The entire homework handout should be completed.
 

Attachments

  • 092208.doc
    68 KB · Views: 29
I don't know if any 4100 students are actually reading this thread but if you are, and you have the text, I would be most grateful if you could copy up the homework assignments from the text and forward them to me.

Thanks in advance.

cdw
 
Wednesday, Sept. 24, 2008

notes attached.

  • Properties of Matrix Operations, continued
    • Properties of Matrix Multiplication
    • Properties of Scalar Multiplication
  • Definition of Vector Space
The homework assignment is from the text:
  • Pages 34-37: 1(a-c,e-g), 2, 3(b,g,h), 4(b,c,f), 5(a,b,d,e,f), 13, 14a, 19, 20, 21(a,b).
  • Pages 48-50: 1, 2, 5, 6a, 7(a,d), 9(a,b), 11, 13, 14 (looks hard but isn't -- can be done in 2 steps), 15, 16, 17, 20a, 29a.
 

Attachments

  • 092408.doc
    70.5 KB · Views: 24
En passant, in a standard text such as Herstein's Topics in Algebra, you'll find a definition of vector space (corresponding to the one in your notes) but also, and more importantly, a host of non-trivial examples that go beyond n-tuples. For example, spaces of functions. A matrix can be considered as a linear operator on elements of a vector space. The set of such linear operators (e.g., matrices) is itself a vector space (in fact, a bit more, as it's both a vector space and a ring, making it an "algebra"). On the space of differentiable functions, the derivative can be considered a linear operator. In short, the terminology of vector spaces subsumes n-tuples and matrices.
 
Mon, Oct 6, 2008

notes attached.

  • Vector Spaces, continued
    • Relating a Ring to a Vector Space
  • Proof that if a system of linear equations has 2 solutions, it has infinite solutions
  • Definition of Linear Algebra
  • Definition of the Transpose of a Matrix
  • Properties of the Transpose
 

Attachments

  • 100608.doc
    141.5 KB · Views: 18
Tue, Oct 14, 2008

notes attached.

  • Transpose of a Matrix, continued
    • Proof of Property 4
  • The Inverse of a Transpose
  • Elementary Row Operations
  • Elementary Matrix
 

Attachments

  • 101408.doc
    121.5 KB · Views: 15
Mon, Oct 20, 2008

notes attached.

  • Proof that, if A is invertible:
    • AX=0 has only the Trivial solution;
    • A can be tranformed into I by a finite series of elementary operations;
    • A can be expressed as a product of elementary matrices.
 

Attachments

  • 102008.doc
    73.5 KB · Views: 16
notes attached.

  • Proof that, if A is invertible:
    • AX=0 has only the Trivial solution;
    • A can be tranformed into I by a finite series of elementary operations;
    • A can be expressed as a product of elementary matrices.

It's finally starting to get interesting. Another way of putting it is that if we consider A to represent a linear mapping between V and itself, then the kernel of this mapping is 0, and the image of the mapping is the entire target vector space V. Invertibility of A implies that A is a bijective (linear) mapping between V and itself; i.e. it does not map V into a subspace of itself.
 
Mon, Oct 27, 2008

notes attached.

  • Diagonal Matrices
  • Triangular Matrices
  • Symmetric Matrices
    • Properties of...
Note: The class for 10/22 was largely a review of 10/20. The class ended with an introduction of Diagonal Matrices which is included in the notes for 10/27.
 

Attachments

  • 102708.doc
    96 KB · Views: 18
Wed, Nov 5, 2008

notes attached.

  • Elementary Products in a Square Matrix
    • Determinants
    • Minor Determinants
    • Cofactors
  • Computing Determinants of Small Matrices
 

Attachments

  • 110508.doc
    61.5 KB · Views: 13
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