/*! 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 3 Rewrite as matrix equations \(A ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 3

Rewrite as matrix equations \(A \mathrm{u}=\mathrm{d}\) (do not solve). The point \(\mathbf{u}=(x, y, 2)\) is on the planes \(x=y, y=z\), \(x-z=1\)

Short Answer

Expert verified
Matrix form: \(A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 0 & -1 \end{bmatrix}\), \(\mathbf{d} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\)."

Step by step solution

01

Identify the variables and the system of equations

The point \(\mathbf{u}=(x, y, 2)\) is subject to the following conditions given by plane equations: 1. \(x = y\)2. \(y = z\)3. \(x - z = 1\).
02

Substitute the value of z

Given that \(z = 2\), substitute \(z\) in the equations:1. \(x = y\)2. \(y = 2\)3. \(x - 2 = 1\).
03

Simplify the equations

From the substituted system, simplify the equations:- From \(y = 2\), we know \(y = 2\).- From \(x - 2 = 1\), we find \(x = 3\).- From \(x = y\), it restates \(x = 3\) and \(y = 2\) satisfy the original systems.
04

Formulate the matrix equation

Express the simplified system in matrix form. The equations can be summarized as:\[ \begin{align*}& 1x - 1y + 0z = 0 \& 0x + 1y - 1z = 0 \& 1x + 0y - 1z = 1 \end{align*} \]This can be rewritten in matrix form \(A\mathbf{u} = \mathbf{d}\) as:\[ A = \begin{bmatrix} 1 & -1 & 0 \ 0 & 1 & -1 \ 1 & 0 & -1 \end{bmatrix}, \quad \mathbf{u} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{d} = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \].

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.

Systems of Equations
When dealing with problems involving systems of equations, you're essentially working with multiple equations that need to be solved together. Each equation in the system shares the same variables, and a solution to the system must satisfy all equations simultaneously. In real-world applications, systems of equations can represent various constraints or conditions.
One method to solve systems of equations is substitution, which involves solving one equation for a variable and using that result in another equation.
Another common method is elimination, which combines equations to cancel out one variable, reducing the number of equations to solve.
Lastly, graphing can offer a visual solution by plotting each equation on a graph and identifying the intersection point, which represents the solution. Each of these methods provides a valuable approach depending on the problem setup and context.
Linear Algebra
Linear Algebra is a branch of mathematics that focuses on vectors, matrix operations, and linear transformations. In the context of this exercise, we're interested in using linear algebra to express systems of equations in matrix form. This greatly simplifies handling complex problems.
Linear algebra involves the study of linear equations, matrices, and vector spaces. Matrices can be used to represent and solve systems of equations, making them an essential tool in various practical applications.
For instance, in engineering, physics, and computer science, solving linear systems efficiently can lead to solutions for optimization problems and computational simulations. By employing matrices, complex systems and transformations become manageable and can be computed efficiently.
Matrix Representation
Matrix Representation is a powerful tool in mathematics, especially when dealing with systems of equations. It allows us to represent a system using matrices, thereby taking advantage of matrix operations to solve the system.
A matrix is essentially a grid of numbers arranged in rows and columns. In the given problem, we saw how the initial system of planes was transformed into a matrix equation, making it easier to handle.
The structure of a matrix form involves the coefficient matrix, variable vector, and a constant matrix. For our problem, this is seen as:
  • The coefficient matrix, where each row represents the coefficients of a variable in one equation.
  • The variable vector reflects the values we're solving for, represented as a column matrix.
  • The constant matrix represents the values on the right-hand side of each equation.
By using this form, large systems of equations can be solved using efficient computational methods, such as Gaussian elimination or matrix inversion.

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) The triple cross product \((A \times B) \times C\) is in the plane of \(\mathbf{A}\) and \(\mathbf{B}\), because it is perpendicular to the cross product _____. (b) Compute \((\mathbf{A} \times \mathbf{B}) \times \mathbf{C}\) when \(\mathbf{A}=a_{2} \mathbf{i}+a_{2} \mathbf{j}+a_{3} \mathbf{k}, \mathbf{B}=\) \(b_{1} \mathbf{i}+b_{2} \mathbf{j}+b_{3} \mathbf{k}, \mathbf{C}=\mathbf{i}\) (c) Compute \((\mathbf{A} \cdot \mathbf{C}) \mathbf{B}-(\mathbf{B} \cdot \mathbf{C}) \mathbf{A}\) when \(\mathbf{C}=\mathbf{i}\). The answers in (b) and (c) should agree. This is also true if \(\mathbf{C}=\mathbf{j}\) or \(\mathbf{C}=\) \(\mathbf{k}\) or \(\mathbf{C}=c_{1} \mathbf{i}+c_{2} \mathbf{j}+c_{3} \mathbf{k} .\) That proves the tricky formula $$(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}=(\mathbf{A} \cdot \mathbf{C}) \mathbf{B}-(\mathbf{B} \cdot \mathbf{C}) \mathbf{A}$$

Describe all points \((x, y)\) such that \(v=x \mathbf{i}+y\) satisfies (a) \(|\mathbf{v}|=2\) (b) \(|v-i|=2\) (c) \(\mathbf{v} \cdot \mathbf{i}=2\) (d) \(\mathbf{v} \cdot \mathbf{i}=|\mathbf{v}|\)

Find \(N\) and the equation of the plane described. Contains the points (2,1,1),(1,2,1),(1,1,2).

Eliminate \(x\) from equation 2 by using equation 1 $$ \begin{aligned} x+2 y+2 z &=0 \\ 2 x+4 y+5 z &=0 \\ 2 y+2 z &=8 \end{aligned} $$ Why can't the new second equation eliminate \(y\) from the third equation? Is there \(a\) solution or is the system singular? Note: If elimination creates a zero in the "pivot position," try to exchange that pivot equation with an equation below it. Elimination succeeds when there is a full set of pivots.

Find the point \((x, y)\) where the two lines intersect (if they do). Ako show how the right side is a combination of the columns on the left side (if it is). ALo find the determinant \(D\). $$ \begin{array}{r} 10 x+y=1 \\ x+y=1 \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.