Linear Algebra True-False 2

1.2.21 1 In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using a different sequence of row operations. January 19, 2017 1/3 1.2.21 1 In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using a different sequence of row operations. False. Theorem 1 says that the RREF is unique. January 19, 2017 1/3 1.2.21 1 In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using a different sequence of row operations. False. Theorem 1 says that the RREF is unique. 2 The row reduction algorithm applies only to augmented matrices for a linear system. January 19, 2017 1/3 1.2.21 1 In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using a different sequence of row operations. False. Theorem 1 says that the RREF is unique. 2 The row reduction algorithm applies only to augmented matrices for a linear system. False. Paragraph two reads: “The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.” January 19, 2017 1/3 1.2.21 1 In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using a different sequence of row operations. False. Theorem 1 says that the RREF is unique. 2 The row reduction algorithm applies only to augmented matrices for a linear system. False. Paragraph two reads: “The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.” 3 A basic variable is a variable that corresponds to a pivot column in the coefficient matrix. January 19, 2017 1/3 1.2.21 1 In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using a different sequence of row operations. False. Theorem 1 says that the RREF is unique. 2 The row reduction algorithm applies only to augmented matrices for a linear system. False. Paragraph two reads: “The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.” 3 A basic variable is a variable that corresponds to a pivot column in the coefficient matrix. True (by definition). Free variables are those that do not correspond to a pivot column. January 19, 2017 1/3 4. Finding a parametric description of the solution set of a linear system is the same as solving the system. January 19, 2017 2/3 4. Finding a parametric description of the solution set of a linear system is the same as solving the system. True. This is one way of describing the set of all solutions. January 19, 2017 2/3 4. Finding a parametric description of the solution set of a linear system is the same as solving the system. True. This is one way of describing the set of all solutions. 5. If one row in the echelon form of an augmented matrix is computed as [0 0 0 5 0], then the associated linear system is inconsistent. January 19, 2017 2/3 4. Finding a parametric description of the solution set of a linear system is the same as solving the system. True. This is one way of describing the set of all solutions. 5. If one row in the echelon form of an augmented matrix is computed as [0 0 0 5 0], then the associated linear system is inconsistent. False. The row corresponds to the variable x4 being equal to zero. The system still may or may not be consistent, depending on what the other rows look like. January 19, 2017 2/3 1.2.22 1 The reduced echelon form of a matrix is unique. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False (question a bit vague, but think about the RREF, or the paragraph after Pivot Positions. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False (question a bit vague, but think about the RREF, or the paragraph after Pivot Positions. 4 A general solution of a system is an explicit description of all solutions of the system. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False (question a bit vague, but think about the RREF, or the paragraph after Pivot Positions. 4 A general solution of a system is an explicit description of all solutions of the system. True (by definition). January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False (question a bit vague, but think about the RREF, or the paragraph after Pivot Positions. 4 A general solution of a system is an explicit description of all solutions of the system. True (by definition). 5 Whenever a system has free variables, the solution set contains many solutions. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False (question a bit vague, but think about the RREF, or the paragraph after Pivot Positions. 4 A general solution of a system is an explicit description of all solutions of the system. True (by definition). 5 Whenever a system has free variables, the solution set contains many solutions. January 19, 2017 3/3 1.2.22 1 The reduced echelon form of a matrix is unique. True. The echelon form is not, but the reduced echelon form is unique (Theorem 1). 2 If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. False. To be consistent, we do NOT want the last column to be a pivot column. 3 The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. False (question a bit vague, but think about the RREF, or the paragraph after Pivot Positions. 4 A general solution of a system is an explicit description of all solutions of the system. True (by definition). 5 Whenever a system has free variables, the solution set contains many solutions. January 19, 2017 3/3