1.
T/F: There exists a field with 37 elements.
Section Daily Prep 4
Today we will discuss finite fields and review linear algebra concepts needed for the remainder of the course. Most of linear algebra works the same over finite fields as over the real numbers, but there are some important differences to keep in mind.
Subsection Learning Objectives
Subsubsection Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
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State the definition and give examples of finite fields and vector spaces over finite fields.
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Perform arithmetic computations in finite fields of prime order \(\Z_p, \F_p, \GF(p)\)
Subsubsection Advanced Learning Objectives
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
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State the definition and give examples of span, linear independence, basis, dimension, rank, row space, null space, generator matrix, parity-check matrix.
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Given a description of a subspace \(S\) of \(\F_p^n\text{,}\) find a basis for \(S\) and write down a generator matrix for \(S\text{.}\)
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Given a description of a subspace \(S\) of \(\F_p^n\text{,}\) find a parity-check matrix for \(S\text{.}\)
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Count the number of subspaces of a given dimension in \(\F_p^n\text{,}\) the number of vectors in a given subspace, and the number of bases for a given subspace.
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
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Guruswami, Rudra, & Sudan, Sections, 2.1-2.2, pp. 31-36 (note that we don’t need the idea of a group right now, so don’t worry about it if you haven’t seen that before)
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Roth, Section 3.1 & Appendix A, pp. 50-51 and 521-526
Exercises Exercises
2.
T/F: There exists a field with 12 elements.
3.
In \(\GF(7)\text{,}\) the field with elements \(\{0,1,2,3,4,5,6\}\) and operations of addition and multiplication modulo 7, what is the result of computing \(2+6\text{?}\)
4.
In \(\GF(7)\text{,}\) the field with elements \(\{0,1,2,3,4,5,6\}\) and operations of addition and multiplication modulo 7, which element is \(-4\text{,}\) i.e. the element which gives \(0\) when added to \(4\text{?}\)
5.
In \(\GF(7)\text{,}\) the field with elements \(\{0,1,2,3,4,5,6\}\) and operations of addition and multiplication modulo 7, which element is \(4^{-1}\text{,}\) i.e. the element which gives \(1\) when multiplied by \(4\text{?}\)
6.
In your own words, how can you check if a subset of \(\F_p^n\) is a subspace?
7.
Decide if each subset of \(\F_3^4\) below is a subspace over \(\F_3\text{.}\)
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\(\displaystyle \{(0000),(1010),(2020)\}\)
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\(\displaystyle \{(0000),(1111),(1010),(2121)\}\)
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\(\displaystyle \{a(1011)+b(0111)\mid a,b\in\F_3\}\)
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The set of all vectors in \(\F_3^4\) with a zero in the first position
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The set of all vectors in \(\F_3^4\) with a one in the first position
8.
What questions do you have about the material for today?
