Linear Algebra TOP Study Guide Questions and CORRECT Answers Is the statement "Two matrices are row equivalent if they have the same number of rows" true or false? Explain. - ✔✔✔ False, because if two matrices are row equivalent it means that t here exists a sequence of row operations that transforms one matrix to the other Is the statement "Elementary row operations on an augmented matrix never change the solution set of the associated linear system" true or false? Explain. - ✔✔✔ True, becaus e the elementary row operations replace a system with an equivalent system. Is the statement "Two equivalent linear systems can have different solution sets" true or false? Explain. - ✔✔✔ False, because two systems are called equivalent if they have th e same solution set. Is the statement "A consistent system of linear equations has one or more solutions" true or false? Explain. - ✔✔✔ True, a consistent system is defined as a system that has at least one solution. Is the statement "Every elementary row operation is reversible" true or false? Explain - ✔✔✔
True, because replacement, interchanging, and scaling are all reversible. Is the statement "A 5×6 matrix has six rows" true or false? Explain. - ✔✔✔ False, because a 5×6 matrix has five rows and six columns. .Is the statement "The solution set of a linear system involving variables x 1...x2 is a list of numbers (s1 ... s2) that makes each equation in the system a true statement when the values s1 ...s2 are subsituted for x1 ... x2 respective ly" true or false? Explain. - ✔✔✔ False, because the description applies to a single solution. The solution set consists of all possible solutions. Is the statement "Two fundamental questions about a linear system involve existence and uniqueness" true or false? Explain. - ✔✔✔ True, because two fundamental questions address whether the solution exists and whether there is only one solution. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequ ences of row operations. Is this statement true or false? - ✔✔✔ The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix. The row reduction algorithm applies only to augmented matrices for a linear system. Is this s tatement true or false? - ✔✔✔ The statement is false. The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. Finding a parametric description of the solution set of a linear system is the same as solving the system. Is this statement true or false? - ✔✔✔ The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution. If one row in an echelon form of an au gmented matrix is [ 0 0 0 5 0] then the associated linear system is inconsistent. Is this statement true or false? - ✔✔✔ The statement is false. The indicated row corresponds to the equation 5x4=0 which does not by itself make the system inconsistent. The equation Ax=b is referred to as a vector equation. Choose the correct answer below. - ✔✔✔ False. The equation Ax=b is referred to as a matrix equation because A is a matrix A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. Choose the correct answer below. - ✔✔✔ True. The equation Ax=b has the same solution set as the equation x1a1 + x2a2 ... + xnan=b The equation Ax=b is consistent if the augmented matrix [A b] has a pivot positio n in every row. Choose the correct answer below - ✔✔✔ False. If the augmented matrix [A b] has a pivot position in every row, the equation Ax=b may or may not be consistent. One pivot position may be in the column representing b The first entry in the pro duct Ax is a sum of products. Choose the correct answer below. - ✔✔✔ True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A If the columns of an m×n matrix A span ℝm, then the equation Ax=b is consistent for each b in ℝm. Choose the correct answer below. - ✔✔✔ True. If the columns of A span ℝm then the equation Ax=b has a solution for each b in ℝm If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm then A cann ot have a pivot position in every row. Choose the correct answer below. - ✔✔✔ True. If A is an m ×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has no solution for some b in ℝm A homogeneous equation is always co nsistent. - ✔✔✔ True. A homogenous equation can be written in the form Ax=0 where A is an m ×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely, x=0. Thus a homogenous equation is always consistent. The equ ation Ax=0 gives an explicit description of its solution set. - ✔✔✔ False. The equation Ax=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set. The homogenous equation Ax= 0 has the trivial solution if and only if the equation has at least one free variable - ✔✔✔ False. The homogeneous equation Ax=0 always has the trivial solution. The equation x=p+tv describes a line through v parallel to p - ✔✔✔ False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation x=p+tv describes a line through p parallel to v. The solution set of Ax=b is the set of all vectors of the form w=p+vn where vn is any solution of the equation Ax=0 - ✔✔✔ False. The solution set could be empty. The statement is only true when the equation Ax=b is consistent for some given b , and there exists a vector p such that p is a solution. Suppose Ax=b has a solution. Explain why the solution is unique precisel y when Ax=0 has only the trivial solution. - ✔✔✔ Since Ax=b is consistent, its solution set is obtained by
True, because replacement, interchanging, and scaling are all reversible. Is the statement "A 5×6 matrix has six rows" true or false? Explain. - ✔✔✔ False, because a 5×6 matrix has five rows and six columns. .Is the statement "The solution set of a linear system involving variables x 1...x2 is a list of numbers (s1 ... s2) that makes each equation in the system a true statement when the values s1 ...s2 are subsituted for x1 ... x2 respective ly" true or false? Explain. - ✔✔✔ False, because the description applies to a single solution. The solution set consists of all possible solutions. Is the statement "Two fundamental questions about a linear system involve existence and uniqueness" true or false? Explain. - ✔✔✔ True, because two fundamental questions address whether the solution exists and whether there is only one solution. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequ ences of row operations. Is this statement true or false? - ✔✔✔ The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix. The row reduction algorithm applies only to augmented matrices for a linear system. Is this s tatement true or false? - ✔✔✔ The statement is false. The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. Finding a parametric description of the solution set of a linear system is the same as solving the system. Is this statement true or false? - ✔✔✔ The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution. If one row in an echelon form of an au gmented matrix is [ 0 0 0 5 0] then the associated linear system is inconsistent. Is this statement true or false? - ✔✔✔ The statement is false. The indicated row corresponds to the equation 5x4=0 which does not by itself make the system inconsistent. The equation Ax=b is referred to as a vector equation. Choose the correct answer below. - ✔✔✔ False. The equation Ax=b is referred to as a matrix equation because A is a matrix A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. Choose the correct answer below. - ✔✔✔ True. The equation Ax=b has the same solution set as the equation x1a1 + x2a2 ... + xnan=b The equation Ax=b is consistent if the augmented matrix [A b] has a pivot positio n in every row. Choose the correct answer below - ✔✔✔ False. If the augmented matrix [A b] has a pivot position in every row, the equation Ax=b may or may not be consistent. One pivot position may be in the column representing b The first entry in the pro duct Ax is a sum of products. Choose the correct answer below. - ✔✔✔ True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A If the columns of an m×n matrix A span ℝm, then the equation Ax=b is consistent for each b in ℝm. Choose the correct answer below. - ✔✔✔ True. If the columns of A span ℝm then the equation Ax=b has a solution for each b in ℝm If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm then A cann ot have a pivot position in every row. Choose the correct answer below. - ✔✔✔ True. If A is an m ×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has no solution for some b in ℝm A homogeneous equation is always co nsistent. - ✔✔✔ True. A homogenous equation can be written in the form Ax=0 where A is an m ×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely, x=0. Thus a homogenous equation is always consistent. The equ ation Ax=0 gives an explicit description of its solution set. - ✔✔✔ False. The equation Ax=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set. The homogenous equation Ax= 0 has the trivial solution if and only if the equation has at least one free variable - ✔✔✔ False. The homogeneous equation Ax=0 always has the trivial solution. The equation x=p+tv describes a line through v parallel to p - ✔✔✔ False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation x=p+tv describes a line through p parallel to v. The solution set of Ax=b is the set of all vectors of the form w=p+vn where vn is any solution of the equation Ax=0 - ✔✔✔ False. The solution set could be empty. The statement is only true when the equation Ax=b is consistent for some given b , and there exists a vector p such that p is a solution. Suppose Ax=b has a solution. Explain why the solution is unique precisel y when Ax=0 has only the trivial solution. - ✔✔✔ Since Ax=b is consistent, its solution set is obtained by
