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Elementary Linear Algebra: An Active Learning Approach

Chapter 1 To Do

To Do: Vectors and Matrices
  1. Find RW examples of matrix mult or matrix-vector mult reasonably interpretable via dpr and / or lc
  2. Use cross-products to find bases for perp spaces of two vectors
  3. Add transpose exercises to Matrix Algebra worksheet
To Do: Determinants (and trace?)
  1. Arrow diagrams for determinant formula.
  2. Trace; include basic theorems. Use to prove that no \(\A,\B\) can satisfy \(\A\B-\B\A=\I\) since \(\text{tr }\left(\A\B-\B\A\right)=0.\)
  3. Get rid of "def" in figure in def of minor
To Do: Vector Spaces
  1. XXXXXXX
To Do: Similarity
  1. Write a section on similarity.
  2. Investigate link between similarity transformations, change-of-basis and Theorem 9.1.15.
  3. Do Heisenberg Uncertainty Principle (see Strang, etc.)
To Do: Projections
  1. Simplify this section a lot; better examples
  2. Fix notation for projections (remove "span" from subscripts?)
  3. Do projections to col\((A)^\perp\) (\(I-P\))
  4. Reverse order of subproblems in Prob 1 in GS hw
To Do: Eigenstuff
  1. More theory in the worksheet for Eigentheory worksheet
  2. Expand on the use of the Cayley-Hamilton Theorem
To Do: Linear Transformations
  1. Fix 11.3 (LTs) #6 to have a solution, create another data set to not have a solution, and create another to not have a solution in a different way.
  2. Prove that the set of linear transformations on \(V\) is itself a vector space.
  3. Improve directions for proving non-linearity for transformations.
  4. Explain why it is useful to use matrices to represent linear transformations.
  5. Replace O.Nan LT example.
  6. Note that for each \(p\in\N,\) left-multiplication by a matrix \(\A\in\R^{m\times n}\) induces a mapping \(T_\A:\R^{n\times p}\,\rightarrow\,\R^{m\times p}.\)
    1. Do various examples
    2. We may consider conditions on \(\A\) for which \(T_\A\) is injective, surjective, both or neither
    3. If \(n=m\) we may consider the case \(p=m\) wherein \(T_\A\) is a mapping on \(\R^{n\times n}.\)
      1. In this case if \(T_\A\) is injective or surjective then \(T_\A\) is bijective. We will prove these facts.
      2. Define identity
      3. Define inverse
      4. Find inverse of diagonal matrices if possible
      5. Show that orthogonal projection is a linear transformation
To Do: Content
  1. Put proof in all theorems even if just a placeholder for worksheet or homework proof ask.
  2. Use term for all terms being defined.
  3. De-Remark many comments, especially those un-referred to.
  4. Worksheet, and edit section for, Least-Squares.
  5. Refresh Logic section.
  6. Create some projects.
  7. Ensure all definitions, theorems, etc. are self-contained. all defined terms
  8. Work the Reading Quizzes.
  9. Create an index.
To Do: Display
  1. Title page including authors.
  2. Fix knowls to display properly or to open a new page (currently displays blank).
  3. Figure out how to toggle the hiding of solutions/proofs.
  4. Fix all “proofs” so they are in the proof environment. Same with Examples.
To Do: Graphics
  1. Find a good 3d plotter for simultaneously plotting planes and vectors.
  2. Figure numbering needs to be modified to avoid gobbling up numbering needed for remarks, definitions, theorems, etc.
  3. Work figures.
  4. Perturb \(A\) and view the \(A^{-1}\) perturbation app (Sage?).
To Do: Worksheets and Homework
  1. Look into putting whitespaces in the worksheets.
  2. Solution manual.
  3. Check all the problems in the worksheets and homeworks are “exercises”.
To Do: Notation
  1. End-of-Proof symbol.
  2. Extensible vector arrows.
  3. Streamline notation for eigenproblem solutions.
To Do: Language and Formatting
  1. Don’t use the word “note” so often.
  2. At the beginning of each section put learning outcomes/expected learning gains including intuition, theory and computation.
  3. Use theorem-like elements (, , , , , , , ).
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