/*! 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 29 Find out whether the given vecto... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

Find out whether the given vectors are dependent or independent; if they are dependent, find a linearly independent subset. In other words, find the dimension of the subspace spanned by the vectors, and a basis for it. Write each of the given vectors as a linear combination of the basis vectors. $$ (3,5,-1),(1,4,2),(-1,0,5),(6,14,5) $$

Short Answer

Expert verified
The given vectors are linearly dependent. The basis is [original]_P

Step by step solution

01

Form the Matrix

Write the given vectors as columns of a matrix:ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline $$$$\begin{aligned}$$$(1)&=&\begin{bmatrix}3 & 1 & -1 & 6\5 & 4 & 0 & 14\-1 & 2 & 5 & 5\breakmatriXEND$$$\begin{aligned}$$$\rightadd,-,-,-,ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline
02

Perform Row Reduction

Convert the matrix to Reduced Row Echelon Form using Gaussian elimination:ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline Step 2: Perform Row ewline ewline ewline Step 2: Perform ewline ewline
03

Determine Pivot Columns

Identify the pivot columns (columns with leading 1s) in the RREF. Corresponding vectors form a basis for the subspace.ewline
04

Write Basis Vectors

From the pivot columns, identify the corresponding original vectors as basis vectors.ewline
05

Write Given Vectors as Linear Combinations

Express each of the original vectors as a linear combination of the basis vectors.ewline ewline$$3(3)-(1)(1)\,14\,

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.

Vector Dependence
In linear algebra, determining whether a set of vectors is dependent or independent is crucial. Vectors are considered linearly dependent if one can be expressed as a linear combination of the others. Conversely, if no vector in the set can be represented as a combination of the others, they are linearly independent.
To identify this, we can form a matrix using the vectors and analyze it through row reduction methods. If any row reduces to zeros, the vectors are dependent. If all rows have leading variables (pivots), the vectors are independent. Understanding this distinction helps in forming a basis for the vector space they span.
Basis
A basis for a vector space is a set of linearly independent vectors that span the entire space. This means any vector in the space can be written as a linear combination of the basis vectors.
For instance, if we have a set of vectors and reduce the corresponding matrix to its reduced row echelon form (RREF), the pivot columns indicate the original vectors that serve as the basis. Finding the basis helps in simplifying complex vector operations and understanding the structure of the vector space.
  • It consists of the minimal number of vectors required to span the space
  • Any vector in the space can be represented using a unique combination of basis vectors
Reduced Row Echelon Form
The Reduced Row Echelon Form (RREF) of a matrix is a unique, simplified version achieved through Gaussian elimination. In RREF, each leading entry (pivot) is 1, and all other entries in the pivot's column are zeros.
To convert a matrix to RREF, follow these steps:
  • Identify the leftmost column that contains a non-zero entry
  • Interchange rows if necessary to move a row with a non-zero entry to the top
  • Normalize the pivot to 1 by row operations
  • Use row operations to ensure all elements in the pivot's column, except the pivot itself, are zeros
This form is especially useful for solving linear equations and is unique for any matrix.
Gaussian Elimination
Gaussian elimination is a method for solving systems of linear equations. It involves a sequence of operations to transform the matrix into either Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).
The steps for Gaussian elimination include:
  • Sequentially eliminate the elements below the pivots to form an upper triangular matrix (REF)
  • Continue, if necessary, to achieve RREF by making all elements above each pivot zero and the pivots themselves one
Gaussian elimination can aid in finding the rank of a matrix, determining the consistency of linear systems, and finding solutions if they exist. It is foundational for understanding and solving linear algebra problems.

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

Show that a real Hermitian matrix is symmetric. Show that a real unitary matrix is orthogonal.

If \(\mathbf{A}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}\) and \(\mathbf{A} \cdot \mathbf{B}=0\), does it follow that \(\mathbf{B}=0:\) (Either prove that it does or give a specific example to show that it doesn't.) Answer the same question if \(\mathrm{A} \times \mathrm{B}=0\). And again answer the same question if \(\mathrm{A} \cdot \mathbf{B}=0\) and \(\mathrm{A} \times \mathbf{B}=0\).

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \begin{aligned} &3 x+y+3 z+6 w=0 \\ &4 x-7 y-3 z+5 w=0 \\ &x+3 y+4 z-3 w=0 \\ &3 x+2 z+7 w=0 \end{aligned} $$

Given the equations $$ \left\\{\begin{array} { l } { x ^ { \prime } = \frac { 1 } { 2 } ( x + y \sqrt { 3 } ) , } \\ { y ^ { \prime } = \frac { 4 } { 2 } ( - x \sqrt { 3 } + y ) } \end{array} \quad \left\\{\begin{array}{l} x^{\prime \prime}=\frac{1}{2}\left(-x^{\prime}+y^{\prime} \sqrt{3}\right) \\ y^{\prime \prime}=-\frac{1}{2}\left(x^{\prime} \sqrt{3}+y^{\prime}\right) \end{array}\right.\right. $$ write each set as a matrix equation and solve for \(x^{n}, y^{\prime \prime}\) in terms of \(x, y\) by multiplying matrices. These equations represent rotations of axes in two dimensions. By comparing them with \((6.3)\) find the rotation angles and check your results.

Write a set of linear equations for each of the following problems and solve them using either determinants or row reduction. An object composed of \(x \mathrm{gm}\) of lead (specific gravity 11\()\) and \(y \mathrm{gm}\) of tin (specific gravity 7 ) weighs \(82 \mathrm{gm}\) in ait and \(77 \mathrm{gm}\) in oil of specific gravity \(\frac{1}{2}\). Find \(x\) and \(y\). .
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

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.