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If the provision + cannot be reformed, it shall be severed from this Public License + without affecting the enforceability of the remaining terms and + conditions. + + c. No term or condition of this Public License will be waived and no + failure to comply consented to unless expressly agreed to by the + Licensor. + + d. Nothing in this Public License constitutes or may be interpreted + as a limitation upon, or waiver of, any privileges and immunities + that apply to the Licensor or You, including from the legal + processes of any jurisdiction or authority. @@ -0,0 +1,29 @@ +# Holden Rohrer's work for the 2021-22 school year + +## Professors + +- Wing Suet Li for MATH 3406: Second Course in Linear Algebra +- Mary Hudachek-Buswell for CS 1332: Data Structures and Algorithms +- Mayya Zhilova for MATH 4261: Math Statistics I +- Martin M Jarrio for PHYS 2212: Intro Physics II + +## Structure + +The growth of this repository will be pretty organic, but as a general +rule it will consist exclusively of ASCII plain-text files (preferably +as a final format, but sometimes using TeX to generate pdfs with a +Makefile). At the top level there will be a folder for each prof, +labeled with their last name (i.e. `markley/`), all-caps files like +README, LICENSE, and PROGRESS for what is mostly metadata, and possibly +some TeX format files. Git commits' commit times will act as the record +for when different events occur. + +## License + +I have chosen to release this writing under CC BY-SA 4.0. +The full license is available in LICENSE. + +The license does not apply to images that are used in +`hireme/final-pres.odp` (which are used under an exception to copyright) +nor any PDF forms I have included (which are likely insufficiently +original to warrant copyright protection). diff --git a/li/01_intro b/li/01_intro new file mode 100644 index 0000000..8e8e9af --- /dev/null +++ b/li/01_intro @@ -0,0 +1,75 @@ +Solution of a system is the set of points/combinations of variables that +satisfy an equation like 3x+2y=1. + +[3 2][x + y] = 1 is equivalent +A := [3 2] +X := [x y]^T +b := 1 +AX = b + +Some systems like: +x - 2y = 0 +x + y = 1 +x - y = 2 +have no solutions. +This can be written as a a matrix too. +Asking "does this system have a solution" is equivalent to asking: +Can [0 1 2]^T be written as a linear combination/in span of [1 1 1]^T +and [-2 1 -1]^T + +----- +How to solve Ax = b where A is an mxn matrix, x is a nx1 matrix of free +variables, and b is a mx1 matrix of constants. + +Ex. +A = [ 3 1 0 4 + 0 1 2 5 + 0 0 -1 1 ] +x = [ x1 x2 x3 x4 ]^T +b = [ 0 1 2 ] + +Find the free variable, and then back-substitute +x4 = t +-x3 + x4 = 2 --> x3 = t - 2 +x2 + 2*x3 + 5*x4 = 1 --> x2 + 2*(t-2) + 5*t = 1 --> x2 = 5 - 7t +3*x1 + x2 + x4 = 0 --> 3*x1 + 5 - 7t + t = 0 --> x1 = -5/3 + 2t + +Pivot element := first non-zero element in a matrix row AND everything +below it is zero + +A matrix is in *row echelon* form iff pivot elements are to the left of +all pivot elements in a lower row. + Therefore, all non-zero rows have a pivot but not every column. +Note: column echelon form also exists. + +Generally, solving Ax = B can be done by solving +CAx = Cb where C is an invertible mxm matrix. +Single row operations used in Gaussian elimination are a subset of these +operations. + +Ex. +A = 3x4 matrix. +Multiply 1/3 to first row: 1/3 R1 -> R1. +C = 3x3 matrix. +[ 1/3 0 0 + 0 1 0 + 0 0 1 ] * A +C^{-1} = +[ 3 0 0 + 0 1 0 + 0 0 1 ] + +Switch rows R2 and R3: +C = [ 1 0 0 + 0 0 1 + 0 1 0 ] +C = C^{-1} + +Add cR1 to R3: +C = [ 1 0 0 + 0 1 0 + c 0 1 ] +C^{-1} = [ 1 0 0 + 0 1 0 + -c 0 1 ] |