This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ith 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 ...
View Full
Document
 Fall '08
 MARKMAN
 Linear Algebra, Algebra, Vectors

Click to edit the document details