TODO.rtf

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\f0\fs24 \cf0 Problems:\
=========\
\
Systems of Linear Equations\
---------------------------\
\
x Augmented matrix of a system of equations\
x Row echelon form\
x Number of solutions of a system of linear equations\
x Pivot columns\
x System of linear equations (1-2)\
\
Applications of Systems of Linear Equations\
-------------------------------------------\
\
x Balancing chemical equations (1-2)\
x Traffic flow (part 1-2)\
  Polynomial interpolation (1-2)\
x Simplified PageRank\
\
Linear combinations\
-------------------\
\
x Computing linear combinations\
x Vector equations (1-2)\
x Interceptor\
\
Span\
----\
\
x Vectors in Span                     \
x In Span and not in Span\
\
Linear Independence\
-------------------\
\
x Linearly independent sets\
\
\
Matrix Equations\
----------------\
\
x Matrix-vector multiplication\
x Matrix equation\
x Matrix equation with a parameter (1-2) (add randomization)\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf2 x Multiple solutions of a matrix equation\cf0 \
\cf2 x Null space and Span\
x Matrices and matrix equations\cf0 \
\
Linear Transformations\
----------------------\
\
\
x House transformations (1-8)\
x Standard matrix from a formula\
x Color mixing\
x Vector with a given value\
x Value of a linear transformation\
x Repeated values of a linear transformation\
x Third vector with the same value\
x One-to-one and onto\
\
Matrix Algebra\
--------------\
\
x Matrix multiplication\
x Product zero\
x Counting paths (1-2)\
x Computing matrix inverses (1-3)\
x Invertible vs non-invertible\
x Solving for matrices\
x Hill cipher\
\
Determinants\
------------\
\
x Determinants of a matrices with parameters (1-6)\
x Determinants and area (parallelogram, triangle, quadrilateral, pentagon)\
x Area and linear transformations (1-2)\
x On the same side\
\
Subspaces\
---------\
\
x Identifying subspaces of R^3\
\
\
Bases\
-----\
\
x Bases of R^3\
x Bases of a null space and a column space\
x Basis of a span\
x Basis or not a basis\
\
Coordinate systems\
------------------\
\
x Vectors and coordinates (1-2) \
x Coordinate change\
\
\
Dimension\
---------\
\
x Dimension of a span\
x Rank and nullity\
x Rank and nullity estimates (1-10)\
\
\
Dot product\
-----------\
 \
x Dot product computations\
\cf2 x Length of a vector\cf0 \
x Distance between vectors\
x Unit vector\
\cf2 x Finding orthogonal vectors\cf3 \
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf0 Orthogonal bases\
----------------\cf3 \
\
\pard\pardeftab720\partightenfactor0
\cf2 x Orthogonal basis coordinates\
x Gram-Schmidt process\cf3 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf0 x Orthogonal basis of a column space\
\
Orthogonal projections\
----------------------\
\
x Projection with an orthogonal basis\
x Orthogonal component\
x Orthogonal vector\
\
Least squares\
-------------\
\
x Least squares via projections\
x Least square solutions of linear equations (1-2)\
x Least square fitting\
\
\
Eigenvalues and eigenvectors\
----------------------------\
\
x Characteristic polynomial\
x Eigenvalue\
x Eigenspace basis\
\
Diagonalization of matrices\
---------------------------\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf2 x Computing diagonalizations (1-3)\cf0 \
x Matrix with given eigenvectors\
\
\
Symmetric matrices\
------------------\
\
  Computing orthogonal diagonalization\
x Missing eigenvector (1-2)\
\
\
Total: 71\
\
\
\
Problem ideas:\
==============\
\
- using signs of determinants to determine if s point is inside or outside a triangle on a plane!!!\
\
- find values of a parameter for which vectors are linearly dependent (Lay 1.7 15-20)\
\
- standard matrix of a linear transformation which takes a give basis to specified values.\
\
- 3x3 matrices for linear transformations in homogeneous coords\
\
\
\
Homework 2018\
=============\
\
\
\cf3 Homework 1\
----------\cf0 \
\
- **Sec. 1.2:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 2  done \
4  done \
8  done \
10 done \
12 done \
14 done \
18 done \
20 done\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 24, 26, 28 theory\cf0 \
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Augmented matrix of a system of equations\
x Row echelon form\
x Number of solutions of a system of linear equations\
x Pivot columns\
x System of linear equations (1-2)\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf0 \
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 Homework 2\
----------\cf0 \
\
- **Sec. 1.6:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 6  done\
8. done\
12 done\
14 done\cf0 \
\
- **Sec. 1.3:** \
\cf4 2  vector algebra\
4. vector algebra\
6. vector algebra\cf0 \
\
\cf4 - :download:`Additional Problem 1 <_static/hw1_additional.pdf>`\
Google PageRank done\cf0 \
\
\cf4 - **Computing assignment 1 (The interceptor)**\cf0 \
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Balancing chemical equations (1-2)\
x Traffic flow (part 1-2)\
  Polynomial interpolation (1-2)\
x Simplified PageRank\
\
x Computing linear combinations\
x Vector equations (1-2)\
x interceptor\cf0 \
\
\
Homework 3\
----------\
\
- **Sec. 1.4:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 2   done\
4   done\cf0 \
\cf4 12  matrix equation DONE\cf0 \
14  check if vector in Span of matrix columns\
16  solutions of a matrix equation\
\
\
- **Sec. 1.7:**  \
\cf4 2   done\
4   done\
6   done  \
8   done\cf0 \
10  Span/lin independence with a parameter\
12  linear independence with a a parameter \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 22, 34, 36, 38  theory \cf0 \
\
\
- **Sec. 4.2:** \
2   check if vector in Nul A\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 6   vectors spanning Nul A DONE\cf0 \
22  given A find a non-zero vector in Col A and a non-hero vector in Nul A    \
24  check if a given vector is in Nul A, Col A\
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Vectors in Span                     \
x In Span and not in Span \
x Linearly independent sets	\
x Matrix-vector multiplication\
x Matrix equation\
x Matrix equation with a parameter (1-2) (add randomization)\
x Null space and Span\
x Matrices and matrix equations\cf0 \
\
Homework 4\
----------\
\
.. rubric:: Due: Thursday, September 27\
\
\
- **Sec. 1.8:** \
2   find images of vectors under a matrix transformation  \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf6 4   inverse image under a linear transformation DONE\cf0 \
8   dimensions of a matrix defining a matrix transformation\
\cf6 10  inverse image under a linear transformation DONE\cf0 \
12  determine if a vector is in the range of a matrix transformation\
14, 16  geometric description of a linear transformation \
20  determine the standard matrix of a linear transformation\
32  show that a function is not a linear transformation\
\
\
- **Sec. 1.9:** \
2   find the standard matrix of a linear transformation\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 16  done \cf0 \
22  image of a vector under a linear transformation\
\cf4 26  done\
38  done\
40. done\cf0 \
\
\cf4 - **Computing assignment 2 (House transformations)**\
- **Computing assignment 3 (Color mixing)**\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 \
x Multiple solutions of a matrix equation\
x House transformations (1-8)\
x Standard matrix from a formula\
x Color mixing\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf7 x Vector with a given value\cf3 \
x Value of a linear transformation\
x Repeated values of a linear transformation\
x Third vector with the same value\
x One-to-one and onto\cf0 \
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 Homework 5\
----------\cf0 \
\
- **Sec. 2.1:** \
2   matrix addition/multiplication\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 10  blah\
12  done\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 16  theory\
18  theory\
22  theory\cf0 \
\
- **Sec. 2.2:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 2   done  \
4   done\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 14  theory\
16  theory\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 30  done\
32  done\cf0 \
\
- **Sec. 2.3:** \
\cf4 4.  done\
6   done\
8   done\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 14  theory\
18  theory\cf0 \
34  formula for the inverse of a linear transformation \
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 - **Computing assignment 4 (Hill ciphers)**\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf0 \
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Matrix multiplication\
x Product zero\
x Path in a network (1-2)\
x Computing matrix inverses (1-3)\
x Invertible vs non-invertible\
x Solving for matrices\
x Hill cipher\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf0 \
\
\
Extra credit 1\
--------------\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 Hill ciphers\cf0 \
\
\
Homework 6\
----------\
\
\
- **Sec. 3.1:** \
2, 10, 14  compute determinants by cofactor   \
  \
     \
- **Sec. 3.2:** \
8   compute determinants by row red\
\cf4 16  dome\
18  done\
20  done\cf0 \
40. dets of products of matrices \
   \
- **Sec. 3.3:** \
\
2, 6 systems of eqs by Cramers method\
12, 16  matrix inverse by Cramers method \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 \
x Determinants of a matrices with parameters (1-6)\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf0 \
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 Homework 7\
----------\cf0 \
\
\
- **Sec. 3.3:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 20   done \
22   done\
28   done\cf0 \
 \
- **Sec. 4.1:** \
\cf4 2, 6, 8, 22  done\cf0 \
\
- **Sec. 4.2:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 30, 36 show that sets of vectors are subspaces\cf0 \
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Determinants and area (parallelogram, triangle, quadrilateral, pentagon)\
x Area and linear transformations (1-2)\
x On the same side\
x Identifying subspaces of R^3\cf0 \
\
Extra credit 2\
--------------\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 - extra_credit2_tutorial.ipynb \cf0 \
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 Homework 8\
----------\cf0 \
\
- **Sec. 4.3:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 2, 6, 8  done\cf0 \
\cf4 10  find basis of a null space DONE\cf0 \
\cf4 14  find basis of null, col space of a matrix DONE\cf0 \
\cf4 16  find basis of a span of vectors DONE\
20  find basis of a span of vectors DONE\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 24  theory\cf0 \
\
- **Sec. 4.4:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 2.  done\
4.  done\
6.  done\
8.  done\cf0 \
14. find vector coords given basis in the v space of polynomials\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 26. theory\
28. theory\cf0 \
\
- **Sec. 4.5:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 10. find dim of a span DONE\
12. find dim of a span DONE\
14. find dim of a null, col space DONE\cf0 \
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Bases of R^3\
x Bases of a null space and a column space\
x Basis of a span\
x Basis or not a basis\
x Dimension of a span\
x Rank and nullity\cf0 \
\cf3 x Rank and nullity estimates (1-10)\
x Vectors and coordinates (1-2) \
x Coordinate change\cf0 \
\
\
\cf3 Homework 9\
----------\cf0 \
\
- **Sec. 6.1:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 10   find the unit vector in the direction of a given vector DONE\cf0 \
\cf4 14.  find distance between vectors. DONE\cf0 \
\cf4 16.  determine if vectors are orthogonal DONE\
18.  determine if vectors are orthogonal DONE\cf0 \
\
\
- **Sec. 6.2:** \
2.  determine if a set is orthogonal\
4.  determine if a set is orthogonal\
6.  determine if a set is orthogonal\
\cf4 8.  find coords of a vector relative to an orthogonal basis DONE\
10. find coords of a vector relative to an orthogonal basis DONE\cf0 \
\
\
- **Sec. 6.3:** \
\cf4 4.  compute the orthogonal projection of a vector given orthogonal basis DONE\
6.  compute the orthogonal projection of a vector given orthogonal basis DONE\cf0 \
\cf4 8.  decompose a vector into a vector os a subspace and a vector orthogonal to the subspace DONE\
10. decompose a vector into a vector os a subspace and a vector orthogonal to the subspace DONE\
14  find the best approximation of a vector by a vector in a subspace DONE\cf0 \
\
\
- **Sec. 6.4:** \
\cf4 2.  use G-S to transform basis into an orthogonal basis DONE\
6.  use G-S to transform basis into an orthogonal basis DONE\cf0 \
\cf4 10. find orthogonal basis of a col space of a matrix DONE\
12. find orthogonal basis of a col space of a matrix DONE\cf0 \
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Dot product computations\cf0 \
\cf3 x Length of a vector\cf0 \
\cf3 x Distance between vectors\cf0 \
\cf3 x Unit vector\
x Finding orthogonal vectors\
\pard\pardeftab720\partightenfactor0
\cf3 X Orthogonal basis coordinates\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Projection with an orthogonal basis\
x Orthogonal component\
x Orthogonal vector\cf0 \
\pard\pardeftab720\partightenfactor0
\cf3 x Gram-Schmidt process\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Orthogonal basis of a column space\cf0 \
\
\
Homework 10\
-----------\
\
.. rubric:: Due: Thursday, November 29\
\
- **Sec. 6.5:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 2.  done\
4.  done\cf0 \
\cf4 10. least square solutions by orthogonal projection DONE\
12. least square solutions by orthogonal projection DONE\cf0 \
\
- **Sec. 6.6:** \
\cf4 least squares fitting of lines and curves\cf0 \
\
- **Sec. 6.7:** \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 2.  blah\cf0 \
4.  compute inner product of polynomials given by evaluations\
22. compute inner product od polynomials given by integration\
\
\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Least squares via projections\cf0 \
\cf3 x Least square solutions of linear equations (1-2)\
x Least square fitting\cf0 \
\
Homework 11\
-----------\
\
.. rubric:: Due: Thursday, December 6\
\
- **Sec. 5.3:** \
2.  use diagonalization to compute a power of a matrix\
4.  use diagonalization to compute an arbitrary power of a matrix\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf4 8.  diagonalize a matrix DONE \
10. diagonalize a matrix DONE\
12. diagonalize a matrix DONE\
16. diagonalize a matrix DONE\cf0 \
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 22. theory \
26. theory\
28. theory\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
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- **Sec. 7.1:** \
8, 10, 12 determine which matrices are orthogonal, if so find the inverse\
18.  orthogonally diagonalize a matrix\
20.  orthogonally diagonalize a matrix\
22.  orthogonally diagonalize a matrix\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 28.  theory\
30.  theory\cf0 \
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\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf3 x Characteristic polynomial\cf8 \
\cf3 x Eigenvalue\
x Eigenspace basis\cf0 \
\cf3 x Matrix with given eigenvectors\cf0 \
\cf3 x Computing diagonalizations (1-3)\cf0 \
\cf3 x Missing eigenvector (1-2) (?)\cf0 \
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Homework 12\
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- **Sec. 7.4:** \
3.  find singular values of a matrix\
7.  find SVD of a matrix\
9.  find SVD of a matrix\
\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\pardirnatural\partightenfactor0
\cf5 17. theory\cf0 \
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}