/*! 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 20 Schreiben Sie die folgende Doppe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 20

Schreiben Sie die folgende Doppelsumme aus: $$ \sum_{3 \leq i \leq 5,2 \leq j \leq 6} a_{i j} $$

Short Answer

Expert verified
The expanded sum is: \( a_{32} + a_{33} + a_{34} + a_{35} + a_{36} + a_{42} + a_{43} + a_{44} + a_{45} + a_{46} + a_{52} + a_{53} + a_{54} + a_{55} + a_{56} \).

Step by step solution

01

Understand the Sum Bounds

The given double summation is \( \sum_{3 \leq i \leq 5, 2 \leq j \leq 6} a_{ij} \). This indicates that we need to evaluate the sum over the indices \( i \) from 3 to 5 and \( j \) from 2 to 6. The variable \( a_{ij} \) denotes an element in a matrix or a sequence where the indices \( i \) and \( j \) are varying within the given ranges.
02

Expand the Outer Sum

Start by expanding the outer summation, which iterates over \( i \). We will write separate sums for each value of \( i \), where \( i = 3, 4, 5 \). For each \( i \), we will then expand the inner sum over \( j \).
03

Expand the Inner Sum for Each i Value

For \( i = 3 \), calculate the inner sum: \( \sum_{j=2}^{6} a_{3j} = a_{32} + a_{33} + a_{34} + a_{35} + a_{36} \).Next for \( i = 4 \), calculate: \( \sum_{j=2}^{6} a_{4j} = a_{42} + a_{43} + a_{44} + a_{45} + a_{46} \).Finally for \( i = 5 \), calculate: \( \sum_{j=2}^{6} a_{5j} = a_{52} + a_{53} + a_{54} + a_{55} + a_{56} \).
04

Combine All Terms

Combine all the expanded terms from each inner sum:\[ a_{32} + a_{33} + a_{34} + a_{35} + a_{36} + a_{42} + a_{43} + a_{44} + a_{45} + a_{46} + a_{52} + a_{53} + a_{54} + a_{55} + a_{56}. \]These 15 terms represent the full expansion of the double summation.

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.

Matrix Notation
To make sense of the double summation, one must first understand matrix notation. Matrices are often used to organize numbers, symbols, or expressions in rows and columns, forming a rectangular array. For example, in matrix notation, the term \( a_{ij} \) refers to an element located in the \( i \)-th row and \( j \)-th column of this array. Each pair of indices \( (i, j) \) pinpoints a specific location in the matrix.
This notation helps in clearly arranging data, especially for computational purposes. In the context of our double summation, each \( a_{ij} \) represents a unique element that we need to sum over, between specific row and column indices. This is crucial for visualization and correctly performing the summation.
Inner-Outer Sum Expansion
A double summation can often seem overwhelming, but breaking it down through inner-outer sum expansion eases the process significantly. Here, we have two sums: the inner sum and the outer sum.
  • The outer sum iterates over the variable \( i \), covering the indices from 3 to 5. This means we handle one row of this conceptual matrix at a time.
  • The inner sum iterates over \( j \), from 2 to 6, computing the sum within a specific row. This involves collecting elements along a row for each fixed \( i \).
When processing the double summation, you start with the outer sum. Choose a fixed value of \( i \) to handle all its corresponding inner sums. This will make the matrix processing much more manageable, allowing you to process column by column within each row.
Summation Bounds
Understanding summation bounds is vital in correctly performing and expanding any summation. In our exercise, bounds indicate the range of indices that we include in our sum calculations.
For the double summation \( \sum_{3 \leq i \leq 5, 2 \leq j \leq 6} a_{ij} \):
  • \( 3 \leq i \leq 5 \) suggests that \( i \) spans from 3 to 5 inclusive, setting the bounds for the outer sum. It means you handle the third, fourth, and fifth \( i \)-indices or rows.
  • \( 2 \leq j \leq 6 \) stipulates that \( j \) spans from 2 to 6 inclusive within each selected \( i \), defining bounds for the inner sum. For each row chosen by \( i \), move along the second to the sixth columns.
Grasping these bounds ensures that you cover every element needed in the matrix exactly once, avoiding oversights or duplications in your sum.

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

Sei \(X\) eine \(n\)-elementige Menge. (a) Geben Sie eine bijektive Abbildung der Menge aller bin盲ren Folgen der L盲nge \(n\) auf die Menge aller Teilmengen von \(X\) an. (b) Geben Sie damit einen neuen Beweis f眉r die Tatsache \(\mid \mathbf{P}(X) \|=2^{n}\) an.

Zeigen Sie den Satz 眉ber die Bijektivit盲t invertierbarer Abbildungen: Sei \(f: X \rightarrow Y\) eine Abbildung. Wenn es eine Abbildung \(f^{\prime}: Y \rightarrow X\) gibt, so dass gilt $$ f \circ f^{\prime}=\operatorname{id}_{Y} \text { und } f^{\prime} \circ f=\mathrm{id}_{X} $$ dann ist \(f\) bijektiv.

Wie viele Folgen \(\left(a_{1}, a_{2}, \ldots, \mathrm{a}_{\mathrm{n}}\right)\) der L盲nge \(n\) gibt es, wenn die einzelnen Folgenglieder \(a_{i}\) jeweils genau \(q\) Werte annehmen k枚nnen?

Eine bin盲re Folge ist eine Folge, deren Elemente nur 0 und 1 sind. Besteht eine solche Folge aus \(n\) Komponenten, so spricht man von einer bin盲ren Folge der L盲nge \(n\). Wie gro \(脽\) ist die Anzahl der bin盲ren Folgen der L盲nge \(n\) ? [Hinweis: Wenn Sie das nicht schnell sehen, schreiben Sie alle bin盲ren Folgen der L盲ngen 2 und 3 auf; dann erhalten Sie eine Vermutung, die Sie dann 鈥瀗ur noch" beweisen m眉ssen.]

(a) Zeigen Sie, dass in der gew枚hnlichen euklidischen Ebene folgendes gilt: Wenn drei Punkte einer Geraden \(g\) den gleichen Abstand zu einer Geraden \(g^{\prime}\) haben, so haben alle Punkte von \(g\) den gleichen Abstand von \(g^{\prime}\). (b) Seien \(g\) und \(g^{\prime}\) zwei Geraden der gew枚hnlichen euklidischen Ebene, die nicht parallel sind. Zeigen Sie, dass es zwei Punkte von \(g\) gibt, die den gleichen \(\mathrm{Ab}\) stand zu \(g^{\prime}\) haben
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Discrete 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.