/*! 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} Q4.5-26Q Let \(H\) be an \(n\)-dimensiona... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Q4.5-26Q (page 191)

Let \(H\) be an \(n\)-dimensional subspace of an \(n\)-dimensional vector space \(V\). Show that \(H = V\).

Short Answer

Expert verified

It is proved that \(H = V\).

Step by step solution

01

State the basis theorem

Let \(V\) be a p-dimensional vector space, such that \(p \ge 1\). Anylinearly independent setof exactly \(p\) elements in \(V\) is automatically a basis for \(V\). Any set of exactly \(p\) elements that spans \(V\) is automatically a basis for \(V\).

02

Show that \(H = V\)

When \(\dim V = \dim H = 0\), \(V = \left\{ 0 \right\}\) and \(H = \left\{ 0 \right\}\), and thus, \(H = V\).

Consider \(\dim V = \dim H > 0\). Then, \(H\) contains a basis \(S\) of \(n\) vectors. However, \(S\) is also a basis for \(V\) according to the basis theorem. Hence, \(H = V = {\mathop{\rm Span}\nolimits} S\).

Thus, it is proved that \(H = V\).

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

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.

In Exercises 7-10, let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},{{\mathop{\rm b}\nolimits} _2}} \right\}\) and \(C = \left\{ {{{\mathop{\rm c}\nolimits} _1},{{\mathop{\rm c}\nolimits} _2}} \right\}\) be bases for \({\mathbb{R}^2}\). In each exercise, find the change-of-coordinates matrix from \(B\) to \(C\) and the change-of-coordinates matrix from \(C\) to \(B\).

10. \({{\mathop{\rm b}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}7\\{ - 2}\end{array}} \right),{{\mathop{\rm b}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right),{{\mathop{\rm c}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}4\\1\end{array}} \right),{{\mathop{\rm c}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}5\\2\end{array}} \right)\).

The first four Hermite polynomials are \(1,2t, - 2 + 4{t^2},\) and \( - 12t + 8{t^3}\). These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of \({{\mathop{\rm P}\nolimits} _3}\).

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

16.

  1. If\(B\)is the standard basis for\({\mathbb{R}^n}\)then the\(B\)-coordinate vector of an\({\mathop{\rm x}\nolimits} \)in\({\mathbb{R}^n}\)is x itself.
  2. The correspondence\({\left( {\mathop{\rm x}\nolimits} \right)_B} \mapsto {\mathop{\rm x}\nolimits} \)is called coordinate mapping.
  3. In some cases, a plane in\({\mathbb{R}^3}\)can be isomorphic to\({\mathbb{R}^2}\).

Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.