Use back substitution to solve each of the following systems of equations: (a) x1 3x2 = 2 2x2 = 6 (b) x1 + x2 + x3 = 8 2x2 + x3 = 5 3x3 = 9 (c) x1 + 2x2 + 2x3 + x4 = 5 3x2 + x3 2x4 = 1 x3 + 2x4 = 1 4x4 = 4 (d) x1 + x2 + x3 + x4 + x5 = 5 2x2 + x3 2x4 + x5 = 1 4x3 + x4 2x5 = 1 x4 3x5 = 0 2x5 = 2
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Table of Contents
1
Matrices and Systems of Equations
1.1
Systems of Linear Equations
1.2
Row Echelon Form
1.3
Matrix Arithmetic
1.4
Matrix Algebra
1.5
Elementary Matrices
1.6
Partitioned Matrices
2
Determinants
2.1
The Determinant of a Matrix
2.2
Properties of Determinants
2.3
Additional Topics and Applications
3
Vector Spaces
3.1
Definition and Examples
3.2
Subspaces
3.3
Linear Independence
3.4
Basis and Dimension
3.5
Change of Basis
3.6
Row Space and Column Space
4
Linear Transformations
4.1
Definition and Examples
4.2
Matrix Representations of Linear Transformations
4.3
Similarity
5
Orthogonality
5.1
The Scalar Product in Rn
5.2
Orthogonal Subspaces
5.3
Least Squares Problems
5.4
Inner Product Spaces
5.5
Orthonormal Sets
5.6
The GramSchmidt Orthogonalization Process
5.7
Orthogonal Polynomials
6
Eigenvalues
6.1
Eigenvalues and Eigenvectors
6.2
Systems of Linear Differential Equations
6.3
Diagonalization
6.4
Hermitian Matrices
6.5
The Singular Value Decomposition
6.6
Quadratic Forms
6.7
Positive Definite Matrices
6.8
Nonnegative Matrices
7
Numerical Linear Algebra
7.1
Floating-Point Numbers
7.2
Gaussian Elimination
7.3
Pivoting Strategies
7.4
Matrix Norms and Condition Numbers
7.5
Orthogonal Transformations
7.6
The Eigenvalue Problem
7.7
Least Squares Problems
Textbook Solutions for Linear Algebra with Applications
Chapter 1.1 Problem 10
Question
Consider a system of the form
\(\begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=0 \\ a_{21} x_{1}+a_{22} x_{2}=0 \end{array}\)
where \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\) are constants. Explain why a system of this form must be consistent.
Solution
Step 1 of 2
Given system of equations is
\(\begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=0 \\ a_{21} x_{1}+a_{22} x_{2}=0 \end{array}\)
Here, \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\) are constants.
We are asked to explain why the system of this form must be consistent.
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full solution
full solution
Title
Linear Algebra with Applications 9
Author
Steven J. Leon
ISBN
9780321962218
. Consider a system of the form a11x1 + a12x2 = 0 a21x1 + a22x2 = 0 where a11, a12, a21
Chapter 1.1 textbook questions
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Chapter 1: Problem 1 Linear Algebra with Applications 9
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Chapter 1: Problem 2 Linear Algebra with Applications 9
Write out the coefficient matrix for each of the systems in Exercise 1.
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Chapter 1: Problem 3 Linear Algebra with Applications 9
In each of the following systems, interpret each equation as a line in the plane. For each system, graph the lines and determine geometrically the number of solutions. (a) x1 + x2 = 4 x1 x2 = 2 (b) x1 + 2x2 = 4 2x1 4x2 = 4 (c) 2x1 x2 = 3 4x1 + 2x2 = 6 (d) x1 + x2 = 1 x1 x2 = 1 x1 + 3x2 = 3
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Chapter 1: Problem 4 Linear Algebra with Applications 9
Write an augmented matrix for each of the systems in Exercise 3.
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Chapter 1: Problem 5 Linear Algebra with Applications 9
Write out the system of equations that corresponds to each of the following augmented matrices: (a) 3 2 8 1 5 7 (b) 5 2 1 3 2 3 4 0 (c) 214 1 4 2 3 4 526 1 (d) 4 312 4 3 1 5 6 5 1124 8 513 2 7
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Chapter 1: Problem 6 Linear Algebra with Applications 9
Solve each of the following systems. (a) x1 2x2 = 5 3x1 + x2 = 1 (b) 2x1 + x2 = 8 4x1 3x2 = 6 (c) 4x1 + 3x2 = 4 2 3 x1 + 4x2 = 3 (d) x1 + 2x2 x3 = 1 2x1 x2 + x3 = 3 x1 + 2x2 + 3x3 = 7 (e) 2x1 + x2 + 3x3 = 1 4x1 + 3x2 + 5x3 = 1 6x1 + 5x2 + 5x3 = 3 (f) 3x1 + 2x2 + x3 = 0 2x1 + x2 x3 = 2 2x1 x2 + 2x3 = 1 (g) 1 3 x1 + 2 3 x2 + 2x3 = 1 x1 + 2x2 + 3 2 x3 = 3 2 1 2 x1 + 2x2 + 12 5 x3 = 1 10 (h) x2 + x3 + x4 = 0 3x1 + 3x3 4x4 = 7 x1 + x2 + x3 + 2x4 = 6 2x1 + 3x2 + x3 + 3x4 = 6
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Chapter 1: Problem 7 Linear Algebra with Applications 9
The two systems 2x1 + x2 = 3 4x1 + 3x2 = 5 and 2x1 + x2 = 1 4x1 + 3x2 = 1 have the same coefficient matrix but different righthand sides. Solve both systems simultaneously by eliminating the first entry in the second row of the augmented matrix 2 1 3 1 4 3 5 1 and then performing back substitutions for each of the columns corresponding to the right-hand sides.
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Chapter 1: Problem 8 Linear Algebra with Applications 9
Solve the two systems x1 + 2x2 2x3 = 1 2x1 + 5x2 + x3 = 9 x1 + 3x2 + 4x3 = 9 x1 + 2x2 2x3 = 9 2x1 + 5x2 + x3 = 9 x1 + 3x2 + 4x3 = 2 by doing elimination on a 3 5 augmented matrix and then performing two back substitutions.
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Chapter 1: Problem 9 Linear Algebra with Applications 9
Given a system of the form m1x1 + x2 = b1 m2x1 + x2 = b2 where m1, m2, b1, and b2 are constants: (a) Show that the system will have a unique solution if m1 = m2. (b) Show that if m1 = m2, then the system will be consistent only if b1 = b2. (c) Give a geometric interpretation of parts (a) and (b).
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Chapter 1: Problem 10 Linear Algebra with Applications 9
. Consider a system of the form a11x1 + a12x2 = 0 a21x1 + a22x2 = 0 where a11, a12, a21, and a22 are constants. Explain why a system of this form must be consistent.
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Chapter 1: Problem 11 Linear Algebra with Applications 9
Give a geometrical interpretation of a linear equation in three unknowns. Give a geometrical description of the possible solution sets for a 3 3 linear system.
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