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Chapter 1: Problem 21

In Exercises 21 and \(22,\) mark each statement True or False. Justify each answer." a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. b. The row reduction algorithm applies only to augmented matrices for a linear system. c. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. d. Finding a parametric description of the solution set of a linear system is the same as solving the system. e. If one row in an echelon form of an augmented matrix is 艗 0 0 0 5 0 , then the associated linear system is inconsistent.

Short Answer

Expert verified
a. False, b. False, c. True, d. True, e. True.

Step by step solution

01

Analyze Statement a

Statement a is: "In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations." The reduced row echelon form (RREF) of a matrix is unique. Therefore, once a matrix is in RREF, it cannot be further reduced into another form of RREF using valid row operations. Thus, this statement is **False**.
02

Analyze Statement b

Statement b is: "The row reduction algorithm applies only to augmented matrices for a linear system." The row reduction algorithm can be applied to any matrix, not just to the augmented matrices of linear systems. We use it to simplify matrices, find ranks, and determine pivots. Hence, this statement is **False**.
03

Analyze Statement c

Statement c is: "A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix." A basic variable is defined as a variable that corresponds to a pivot column in a matrix during the row reduction process. This definition is correct, so this statement is **True**.
04

Analyze Statement d

Statement d is: "Finding a parametric description of the solution set of a linear system is the same as solving the system." Solving a linear system involves deriving a form that expresses all solutions, if they exist. A parametric description offers a complete characterization of the solution set, often involving parameters for free variables. Therefore, this statement is **True**.
05

Analyze Statement e

Statement e is: "If one row in an echelon form of an augmented matrix is 艗 0 0 0 5 0 , then the associated linear system is inconsistent." In this row, the existence of a non-zero constant (5) without corresponding variables signifies a contradiction ( 0 = 5 ). Therefore, the linear system is indeed inconsistent. Hence, this statement is **True**.

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Key Concepts

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

Row Reduction
Row reduction is a fundamental technique in linear algebra, often used to simplify matrices into more manageable forms. The process involves a sequence of operations known as Gaussian elimination, which transforms a matrix into an echelon form or the more simplified reduced row echelon form (RREF). These operations include:
  • Swapping two rows.
  • Multiplying a row by a non-zero scalar.
  • Adding or subtracting the multiple of one row to another row.
This method is not limited to matrices from linear systems; it is a tool widely used for various matrix-related tasks like finding determinants or computing inverses. It's crucial to remember that while the initial stages may yield multiple equivalent matrices, when reduced to RREF, the form is unique.
Reduced Row Echelon Form (RREF)
The Reduced Row Echelon Form (RREF) of a matrix is a special type of echelon form that takes simplification a step further. In RREF, every leading entry (also known as a pivot) is 1, and is the only non-zero entry in its column. Additionally, each pivot is to the right of all the pivots in the rows above it.
RREF provides a clear path to understanding the solutions of a linear system. Because the form is unique, it ensures that any solutions derived from this matrix are consistent, assuming the system is consistent. No matter the sequence of row operations, any matrix, once in RREF, will look the same.
Pivot Columns
In the context of matrices, pivot columns are crucial as they determine the basic structure of the solution to a system of linear equations. A pivot column is any column in a matrix that contains a pivot position 鈥� the position of the leading 1 in a row. Pivot columns are associated with basic variables, which essentially form the backbone of the solution to the system. Basic variables correspond to these columns, indicating that they are determined precisely by the system of equations. The remaining non-pivot columns, typically associated with free variables, contribute the parameters to the solution.
Parametric Description
A parametric description is an expression of the solution set of a linear system where the free variables are treated as parameters. This kind of description is pivotal for systems that have infinitely many solutions. In finding a parametric description, the basic variables are expressed in terms of the free variables, which can take on any value. This capture of the entire solution space through parameters is akin to solving the linear system. It provides a complete and versatile framework to understand how solutions vary with changes in the free variables, giving insight into the nature of the solutions and how they relate to each other.

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Most popular questions from this chapter

Suppose an economy has only two sectors, Goods and Services. Each year, Goods sells 80\(\%\) of its output to Services and keeps the rest, while Services sells 70\(\%\) of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures.

Determine by inspection whether the vectors are linearly independent. Justify each answer. \(\left[\begin{array}{l}{4} \\\ {4}\end{array}\right],\left[\begin{array}{r}{-1} \\\ {3}\end{array}\right],\left[\begin{array}{l}{2} \\\ {5}\end{array}\right],\left[\begin{array}{l}{8} \\ {1}\end{array}\right]\)

In Exercises \(17-20,\) show that \(T\) is a linear transformation by finding a matrix that implements the mapping. Note that \(x_{1}, x_{2}, \ldots\) are not vectors but are entries in vectors. $$ T\left(x_{1}, x_{2}\right)=\left(2 x_{2}-3 x_{1}, x_{1}-4 x_{2}, 0, x_{2}\right) $$

[M] In Exercises \(37-40\) , determine if the columns of the matrix span \(\mathbb{R}^{4} .\) $$ \left[\begin{array}{rrrrr}{12} & {-7} & {11} & {-9} & {5} \\ {-9} & {4} & {-8} & {7} & {-3} \\ {-6} & {11} & {-7} & {3} & {-9} \\ {4} & {-6} & {10} & {-5} & {12}\end{array}\right] $$

Should be solved without performing row operations. [Hint: Write \(A \mathbf{x}=\mathbf{0}\) as a vector equation. Given \(A=\left[\begin{array}{rrr}{4} & {1} & {6} \\ {-7} & {5} & {3} \\ {9} & {-3} & {3}\end{array}\right],\) observe that the first column. plus twice the second column equals the third column. Find a nontrivial solution of \(A \mathbf{x}=\mathbf{0} .\)
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