Term
| Every elementary row operation is reversible. |
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Definition
| True. Because if you simple do the inverse of the initial operation you go back to the original row. |
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Term
| A 5x6 matrix has six rows. |
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Definition
| False. A 5x6 matrix has five rows. |
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Term
| The solution set of linear system involving variables x1,...,xn is a list of numbers (s1,...,sn) that makes each equation in the system a true statement when the values s1,...,sn are substituted for x1,...,xn, respectively. |
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Definition
| False. The description given applied to a single solution. The solution set consist of all possible solutions. Only in special cases does the solution set consist of exactly one solution. Mark the statement true only if it is always true. |
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Term
| Two fundamental questions about a linear system involve existence and uniqueness. |
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Definition
| True. 1.)Is the system consistent; that is, does at least one solution exist? 2.)If a solution exists, is it the only one; that is, is the solution unique? |
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Term
| Elementary row operations on an augmented matrix never change the solution set of the associated linear system. |
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Definition
| True. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. |
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Term
| Two Matrices are row equivalent if they have the same number of rows. |
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Definition
| False. The definition of row equivalent requires that there exist a sequence of row operations that transforms one matrix into the other. |
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Term
| An inconsistent system has more than one solution. |
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Definition
| False. By definition an inconsistent system has no solution. |
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Term
| Two linear systems are equivalent if they have the same solution set. |
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Definition
| True. Equivalent systems - Linear systems with the same solution set. |
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Term
| In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. |
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Definition
| False. Each matrix is row equivalent to one and only one reduced echelon matrix. |
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Term
| The row reduction algorithm applies only to augmented matrices for a linear system. |
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Definition
| False. It applies to all matrices augmented or not. |
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Term
| A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. |
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Definition
| True. Basic variables are variables that correspond to a pivot column in the coefficient matrix. |
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Term
| Finding a parametric description of the solution set of a linear system is the same as solving the system. |
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Definition
| True. Solving a system amounts to finding a parametric description of the solution set, or determining if it is empty. |
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Term
| If one row in a echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent. |
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Definition
| False. The row shown corresponds to the 5*x4 = 0, which by itself does not lead to a contradiction. So the system might be consistent or it might be inconsistent. |
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Term
| The echelon form of a matrix is unique. |
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Definition
| False. Only the reduced echelon form is unique. |
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Term
| The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. |
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Definition
| False. The pivot positions in the matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix. |
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Term
| Reducing a matrix to echelon form is called the forward phase of the row reduction process. |
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Definition
| True. Forward phase is the row reduction algorithm. |
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Term
| Whenever a system has free variables, the solution set contains many solutions. |
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Definition
| False. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty. |
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Term
| A general solution of a system is an explicit description of all solutions of the system. |
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Definition
| True. The general solution of the system gives an explicit description of all solutions. |
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Term
| Suppose a 3x5 coefficient matrix for a system has three pivot columns. Is the system consistent? Why or Why not? |
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Definition
| Yes. The system is consistent because with three pivots, there must be a pivot in the third (bottom) row of the coefficent matrix. REF cannot have 0 = 1 or 1 = 0. |
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Term
| Suppose a system of linear equations has a 3x5 augmented matrix whose fifth column is a pivot column. Is the system consistent? Why or why not? |
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Definition
| The system is inconsistent because the pivot column 5 means that there is a row of the form [0 0 0 0 1]. Since this is the augmented matrix, Theorem 2 shows that the system has no solution. |
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Term
| Suppose the coefficient matrix of a system of linear equations has a pivot in each column. Explain why the system has a unique solution. |
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Definition
| If the coefficient matrix has a pivot position in every row, then there is a pivot position in the bottom rown, and there is no room for a pivot in the augmented column. So, the system in consistent by Theorem 2. |
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Term
| Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution. |
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Definition
| Since there are three pivots, one in each row, the augmented matrix must reduce to the form of the (identity style) matrix. No matter what is on the other side of the equal sign, the solution exists and is unique. |
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Term
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Definition
| If a linear system is consistent, then the solution is unique if and only if every column in the coefficient matrix is a pivot column. otherwise there are infinitely many solutions. |
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Term
| What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution. |
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Definition
| Every column in the augmented matrix except the rightmost column is the pivot column, and the rightmost column is not a pivot column. |
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