# Coding the matrix. Linear algebra through computer science by Philip N. Klein

By Philip N. Klein

A fascinating creation to vectors and matrices and the algorithms that function on them, meant for the coed who is aware tips to application. Mathematical ideas and computational difficulties are encouraged through functions in laptop technological know-how. The reader learns by means of *doing*, writing courses to enforce the mathematical techniques and utilizing them to hold out projects and discover the purposes. Examples comprise: error-correcting codes, differences in photographs, face detection, encryption and secret-sharing, integer factoring, removal standpoint from a picture, PageRank (Google's score algorithm), and melanoma detection from cellphone gains. A significant other website,

codingthematrix.com

offers info and aid code. many of the assignments should be auto-graded on-line. Over 2 hundred illustrations, together with a range of appropriate *xkcd* comics.

Chapters: *The Function*, *The Field*, *The Vector*, *The Vector Space*, *The Matrix*, *The Basis*, *Dimension*, *Gaussian Elimination*, *The internal Product*, *Special Bases*, *The Singular worth Decomposition*, *The Eigenvector*, *The Linear Program*

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X). In writing Python code, we emphasize the use of comprehensions, which allow one to express computations over the elements of a set, list, or dictionary without a traditional for-loop. Use of comprehensions leads to more compact and more readable code, code that more clearly expresses the mathematical idea behind the computation being expressed. Comprehensions might be new to even some readers who are familiar with Python, and we encourage those readers to at least skim the material on this topic.

This shows g ◦ f is the identity function. We also claim that f ◦ g is the identity function on g’s domain. Let yˆ be any element of g’s domain. By the definition of g, f (g(ˆ y )) = yˆ. 19: Every function has at most one functional inverse. Proof Let f : U → V be an invertible function. Suppose that g1 and g2 are inverses of f . We show that, for every element v ∈ V , g1 (v) = g2 (v), so g1 and g2 are the same function. Let v ∈ V be any element of the co-domain of f . 17), there is some element u ∈ U such that v = f (u).

THE FUNCTION 32 >>> L[5:] [50, 60, 70, 80, 90] Slices that skip You can use a colon-separated triple a:b:c if you want the slice to include every cth element. For example, here is how you can extract from L the list consisting of even-indexed elements and the list consisting of odd-indexed elements: >>> L[::2] [0, 20, 40, 60, 80] >>> L[1::2] [10, 30, 50, 70, 90] Obtaining elements of a list by unpacking The second way to obtain individual elements is by unpacking. Instead of assigning a list to a single variable as in mylist =[4,8,12], one can assign to a list of variables: >>> [x,y,z] = [4*1, 4*2, 4*3] >>> x 4 >>> y 8 I called the left-hand side of the assignment a “list of variables,” but beware: this is a notational fiction.