/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Let \(S=\left\\{u_{1}, u_{2}, \l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 11

Let \(S=\left\\{u_{1}, u_{2}, \ldots, u_{n}\right\\}\) be a linearly independent subset of a vector space \(\mathrm{V}\) over the field \(Z_{2}\). How many vectors are there in \(\operatorname{span}(S)\) ? Justify your answer.

Short Answer

Expert verified
There are \(2^n - 1\) vectors in the span of S.

Step by step solution

01

Understand the Concepts

In this problem, we are working with a linearly independent subset within a vector space over the field Z_2. Linearly independent means that no vector in the set S can be expressed as a linear combination of the others, and Z_2 is a field with only two elements: 0 and 1.
02

Linear Combinations

Since the set S is linearly independent, every vector in the span of S can be written as a unique linear combination of the vectors in S. A linear combination of the vectors in S is of the form: \(c_1u_1 + c_2u_2 + \cdots + c_nu_n\) where \(c_1, c_2, \ldots , c_n\) are scalar coefficients from the field Z_2.
03

Determine the Possible Coefficient Values

Since we are working with the field Z_2, each coefficient can only have two possible values: 0 or 1. That means for each of the n vectors in S, there are 2 possible values for the scalar coefficients.
04

Calculate the Total Number of Vectors in the Span of S

To find the total number of vectors in the span of S, we can multiply the number of possible values for each coefficient: Number of Vectors = \(2^n\) However, one of these combinations corresponds to the zero vector (when all coefficients are 0), which we need to subtract from the total count: Number of Vectors in the span of S = \(2^n - 1\) So, there are \(2^n - 1\) vectors in the span of S.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(F_{1}\) and \(F_{2}\) be fields. A function $g \in \mathcal{F}\left(F_{1}, F_{2}\right)\( is called an even function if \)g(-t)=g(t)\( for each \)t \in F_{1}\( and is called an odd function if \)g(-t)=-g(t)\( for each \)t \in F_{1}$. Prove that the set of all even functions in $\mathcal{F}\left(F_{1}, F_{2}\right)\( and the set of all odd functions in \)\mathcal{F}\left(F_{1}, F_{2}\right)\( are subspaces of \)\mathcal{F}\left(F_{1}, F_{2}\right)$.

Exercises 5 and 6 show why the definitions of matrix addition and scalar multiplication (as defined in Example 2) are the appropriate ones. Richard Gard ("Effects of Beaver on Trout in Sagehen Creek, California," J. Wildlife Management, 25, 221-242) reports the following number of trout having crossed beaver dams in Sagehen Creek. Upstream Crossings $$ \begin{array}{lccc} \hline & \text { Fall } & \text { Spring } & \text { Summer } \\ \hline \text { Brook trout } & 8 & 3 & 1 \\ \text { Rainbow trout } & 3 & 0 & 0 \\ \text { Brown trout } & 3 & 0 & 0 \\ \hline \end{array} $$ Upstream Crossings $$ \begin{array}{lccc} \hline & \text { Fall } & \text { Spring } & \text { Summer } \\ \hline \text { Brook trout } & 9 & 1 & 4 \\ \text { Rainbow trout } & 3 & 0 & 0 \\ \text { Brown trout } & 1 & 1 & 0 \\ \hline \end{array} $$Record the upstream and downstream crossings in two 3 x 3 matrices, and verify that the sum of these matrices gives the total number of crossings (both upstream and downstream) categorized by trout species and season

Find the equation of the plane \(H:\) (a) with normal \(\mathbf{N}=3 \mathbf{i}-4 \mathbf{j}+5 \mathbf{k}\) and containing the point \(P(1,2,-3)\) (b) parallel to \(4 x+3 y-2 z=11\) and containing the point \(Q(2,-1,3)\)

Prove: For any complex numbers \(z, w \in \mathbf{C},|z w|=|z||w|\)

Prove that a set \(S\) of vectors is linearly independent if and only if each finite subset of \(S\) is linearly independent.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.