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By Ladislav Nebeský

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Group Work 2: Beyond Parity You may want to do Part 1 with the students, or a different example of your own devising. Answers 1. [1, 0, 1, 1, 1, 1, 0] 2. [1, 1, 1, 0, 1, 0, 1] 3. If the intended message was [1, 1, 0, 1] then the transmitted word should be [1, 1, 0, 1, 1, 0, 1]. We didn’t get that, so there is an error. Now, since we are assuming there was only one error, we can try flipping each bit of the received word, in turn: [0, 1, 0, 1, 0, 1, 1] is valid [1, 0, 0, 1, 0, 1, 1] is valid [1, 1, 1, 1, 0, 1, 1] is invalid [1, 1, 0, 0, 0, 1, 1] is invalid [1, 1, 0, 1, 1, 1, 1] is invalid [1, 1, 0, 1, 0, 0, 1] is invalid [1, 1, 0, 1, 0, 1, 0] is invalid So we know that the first or second bit is faulty, but we don’t know which.

Points to Stress 1. Definitions of linear dependence and linear independence. 2. The equivalent definitions of linear dependence, specifically Theorems 2–4. 3. Definition of the span of a set of vectors, including geometric interpretation and spanning sets of Rn , including the standard unit vectors. 4. Theorem 1: A system of linear equations A b is consistent if and only if b is a linear combination of the columns of A. Drill Question Consider the following four vectors in R3 . ⎡ ⎤ 1 ⎢ ⎥ ⎣2⎦ 3 Answer Are they linearly independent?

What extra steps are needed to put your matrix into this form? 2 Direct Methods for Solving Linear Systems Suggested Time and Emphasis 1–2 classes. Essential material (systems over Zp optional). Points to Stress 1. Solving systems using elementary row operations (Gaussian elimination and Gauss-Jordan elimination). 2. Row equivalence, row echelon form and the definition and uniqueness of reduced row echelon form 3. The rank of a matrix, including the rank theorem. 4. Homogeneous linear systems. Drill Question Which of the following matrices are in row echelon form?

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Algebraic properties of trees (Acta Universitatis Carolinae : Philologica monographia) by Ladislav Nebeský

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