235h5 - Math 235 Assignment 5 Due October 19, 2011 1: This...

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Unformatted text preview: Math 235 Assignment 5 Due October 19, 2011 1: This question concerns coding theory. We restate the notation from class. Let F be the field with two elements. Let 1001011 H = 0 1 0 1 1 0 1 . 0011110 The kernel of H is spanned by four vectors of the form x x x x x x x x x x x x 1 , v2 = 0 , v3 = 0 , v4 = 0 . v1 = 0 1 0 0 0 0 1 0 0 0 0 1 1a: In the above the x’s are zeros or ones depending on their position. They are not all the same. Find explicitly the elements v1 , v2 , v3 , v4 so that the last four entries are as above and the vi are in the kernel of H . Let M be the matrix of 7 × 4 whose column vectors are v1 , v2 , v3 , v4 . size 0 1 0 1b: Verify explicitly that 1 is in the kernel of the matrix H by performing matrix 0 1 0 multiplication in F. x1 x2 1c: To encode means to take a vector x = ∈ F4 and multiply it by M to obtain the x3 x4 1 1 coded version of x in F7 . What is the code for x = . 0 0 0 1 1 1d: The vector y = 1 is the result of a transmission error introduced into a coded 0 1 0 vector. It is of the form y = M x + ei for x ∈ F4 and ei equal to a vector in F7 with all zero 1 entries except in the i-th position. Apply H to y = M x + ei . Why do you obtain Hei ? Explicitly compute Hy . What is i? 2a: Assume that w ∈ R5 is not in the span of the independent set S = {v1 , v2 , v3 } ⊂ R5 . Is the set {w, v1 , v2 , v3 } independent? Why? 2b: Let v1 , v2 , v3 be an independent set of vectors in R3 . Let M be the matrix whose columns are v1 , v2 , v3 . What is the RREF of M ? Explain. 2 ...
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235h5 - Math 235 Assignment 5 Due October 19, 2011 1: This...

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