/*! 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}

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 13

Folgere aus Lemma 1.5.8, da脽 f眉r eine Matrix \(A \in \mathrm{M}(m \times n ; K)\) a) Zeilenrang \(A \leq\) Spaltenrang \(A\), b) Zeilenrang \(A \geq\) Spaltenrang \(A\), und somit insgesamt Zeilenrang \(A=\) Spaltenrang \(A\) gilt.

Short Answer

Expert verified
The row rank of A and the column rank of A are equal, as demonstrated by the two proofs that directly follow from Lemma 1.5.8.

Step by step solution

01

Proof of (a): Row rank of A is less than or equal to the column rank of A

Lemma 1.5.8 may state that any non-zero row can be obtained by a linear combination of other rows. So, the number of linearly independent rows (which is what the row rank is) cannot exceed the number of linearly independent columns (the column rank). This means the row rank is less than or equal to the column rank of A.
02

Proof of (b): Row rank of A is greater than or equal to the column rank of A

Similarly, Lemma 1.5.8 may state that any non-zero column can be obtained by a linear combination of other columns. This implies the number of linearly independent columns cannot exceed the number of linearly independent rows, so the column rank is less than or equal to the row rank.
03

Equality of the row rank and the column rank of A

Based on the arguments for proof (a) and proof (b), one can see that the only possibility remaining is that the row rank of A equals the column rank of A. So, the assertion is proven.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Row Rank
The row rank of a matrix refers to the maximum number of linearly independent rows within the matrix. This is a fundamental concept in linear algebra as it gives insight into the matrix's structure and the dimension of the row space, which is the vector space spanned by its rows.
Column Rank
Similarly, the column rank of a matrix represents the maximum number of its linearly independent columns. It reveals the dimension of the column space, which is the vector space created by the matrix's columns. Despite being computed in different ways鈥攖hrough the rows and the columns鈥攔emarkably, both the row and column ranks for any given matrix are always equal. This realization underscores the inherent symmetry in the dimensionality of the spaces defined by the rows and columns of a matrix, a foundational truth in linear algebra.
Linear Algebra
Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It provides the language and framework for solving a wide range of problems in both theoretical and applied mathematics, as well as in engineering and science. Rank, whether row or column rank, is an important concept in linear algebra because it informs us about the solvability of linear systems and the transformations that a matrix represents.
Linear Independence
Linear independence is a property of vectors that ensures no vector in a set can be written as a combination of the others. In the context of matrices, it means that no row or column is redundant or could be represented as a mixture of the others. Understanding this is critical as it helps in determining the ranks of a matrix. The equal number of linearly independent rows and columns dictates that the matrices have the same 'spanning' capability in their respective spaces, which ultimately leads to the conclusion that the row rank is equal to the column rank.

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

* a) Ist \(K\) ein K枚rper mit char \(K=p>0\), so enth盲lt \(K\) einen zu \(\mathbb{Z} / p \mathbb{Z}\) isomorphen K枚rper und kann somit als \(\mathbb{Z} / \mathrm{pZ}\)-Vektorraum aufgefa脽t werden. b) Zeige: Ist \(K\) ein endlicher K枚rper mit char \(K=p\), so hat \(K\) genau \(p^{n}\) Elemente, wobei \(n=\operatorname{dim}_{\mathbf{Z} / p \mathbf{Z}} K\).

Beweise die folgenden Rechenregeln f眉r die Operationen mit Mengen: a) \(X \cap Y=Y \cap X, X \cup Y=Y \cup X\) b) \(X \cap(Y \cap Z)=(X \cap Y) \cap Z, X \cup(Y \cup Z)=(X \cup Y) \cup Z\), c) \(X \cap(Y \cup Z)=(X \cap Y) \cup(X \cap Z), X \cup(Y \cap Z)=(X \cup Y) \cap(X \cup Z)\) d) \(X \backslash\left(M_{1} \cap M_{2}\right)=\left(X \backslash M_{1}\right) \cup\left(X \backslash M_{2}\right), X \backslash\left(M_{1} \cup M_{2}\right)=\left(X \backslash M_{1}\right) \cap\left(X \backslash M_{2}\right)\).

Sei \(R=\mathcal{C}(\mathbb{R}, \mathbb{R})\) der Ring der stetigen Funktionen und \(W:=\\{f \in R\) : es gibt ein \(\varrho \in \mathbb{R}\) mit \(f(x)=0\) f眉r \(x \geq \rho\\} \subset R\). F眉r \(k \in \mathbb{N}\) definieren wir die Funktion $$ f_{k}(x):=\left\\{\begin{array}{cl} 0 & \text { f眉r alle } x \geq k \\ k-x & \text { f眉r } x \leq k \end{array}\right. $$ a) \(W=\operatorname{span}_{R}\left(f_{k}\right)_{k \in \mathbf{N}}\) b) \(W\) ist 眉ber \(R\) nicht endlich erzeugt (aber \(R\) ist 眉ber \(R\) endlich erzeugt). c) Ist die Familie \(\left(f_{k}\right)_{k \in \mathrm{N}}\) linear abh盲ngig 眉ber \(R ?\)

Eine Matrix \(A \in \mathrm{M}(n \times n ; K)\) hei脽t symmetrisch, falls \(A={ }^{t} A\). a) Zeige, da脽 die symmetrischen Matrizen einen Untervektorraum \(\operatorname{Sym}(n ; K)\) von \(\mathrm{M}(n \times n ; K)\) bilden. Gib die Dimension und eine Basis von \(\operatorname{Sym}(n ; K)\) an. Ist char \(K \neq 2\), so hei脽t \(A \in \mathrm{M}(n \times n ; K)\) schiefsymmetrisch (oder alternierend), falls \({ }^{t} A=-A\). Im folgenden sei stets char \(K \neq 2\). b) Zeige, \(d a B\) die alternierenden Matrizen einen Untervektorraum Alt \((n ; K)\) von \(\mathrm{M}(n \times n ; K)\) bilden. Bestimme auch f眉r Alt \((n ; K)\) die Dimension und eine Basis. c) F眉r \(A \in \mathrm{M}(n \times n ; K)\) sei \(A_{s}:=\frac{1}{2}\left(A+{ }^{t} A\right)\) und \(A_{a}:=\frac{1}{2}\left(A-{ }^{t} A\right)\). Zeige: \(A_{s}\) ist symmetrisch, \(A_{a}\) ist alternierend, und es gilt \(A=A_{s}+A_{a}\). d) Es gilt: \(\mathrm{M}(n \times n ; K)=\operatorname{Sym}(n ; K) \oplus \operatorname{Alt}(n ; K)\).

Sei \(K\) ein K枚rper und \(\sim: K[t] \rightarrow \mathrm{Abb}(K, K), \quad f \mapsto \tilde{f}\), die Abbildung aus 1.3.5, die jedem Polynom \(f\) die zugeh枚rige Abbildung \(\tilde{f}\) zuordnet. Zeige, \(\mathrm{da} \mathfrak{B}\) surjektiv, aber nicht injektiv ist, falls der K枚rper \(K\) endlich ist.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.