Section 1.2 21

I In some cases a matrix may be row reduced to more than onematrix in reduced row echelon form, using different sequencesof row operations. FALSE

I The row reduction algorithm applies only to augmentedmatrices for a linear system. FALSE

I A basic variable in a linear system is a variable thatcorresponds to a pivot column in the coefficient matrix. TRUE

I Finding a parametric description of the solution set of a linearsystem is the same as solving the system. TRUE

I If one row in an echelon form of an augmented matrix is[0 0 0 5 0 ], then the associated linear system is inconsistent.FALSE

Linear Algebra, David Lay Week Two True or False

Section 1.2 22

I The echelon form of a matrix is unique FALSE

I The pivot positions in a matrix depend on whether rowinterchanges are used n the row reduction process. FALSE

I Reducing a matrix to echelon form is called the forward phaseof the row reduction process. TRUE

I Whenever a system has free variables, the solution setcontains many solutions. FALSE

I A general solution os a system is an explicit description os allsolutions of the system. TRUE

Linear Algebra, David Lay Week Two True or False

Section 1.3 23

I Another notation for the vector

[−43

]is

[−4 3

]. FALSE

I The points in the place corresponding to

[−25

]and

[−52

]lie on a line through the origin. FALSE

I An example of a linear combination of vectors v1 and v2 is thevector 1/2v1 TRUE

I The solution set of the linear system whose augmented matrixis [a1 a2 a3 b] is the same as the solution set of the equationx1a1 + x2a2 + x3a3 = b TRUE

I The set Span {u, v} is always visualized as a plane throughthe origin. FALSE

Linear Algebra, David Lay Week Two True or False

Section 1.3 24

I Any list of five real numbers is a vector in R5. TRUE

I The vector u results when a vector u − v is added to thevector v . TRUE

I The weights c1, . . . , cp is a linear combinationc1v1 + · · ·+ cpvp cannot all be zero. FALSE

I When u and v are nonzero vectors, Span {u, v} contains theline through u and the origin. TRUE

I Asking whether the linear system corresponding to anaugmented matrix [a1 a2 a3 b] has a solution amounts toasking whether b is in Span {a1, a2, a3}. TRUE

Linear Algebra, David Lay Week Two True or False

Section 1.4 23

I The equation Ax = b is referred to as the vector equation.FALSE

I The vector b is a linear combination of the columns of amatrix A if and only if the equation Ax = b has at least onesolution. TRUE

I The equation Ax = b is consistent if the augmented matrix[A b] has a pivot position in every row. FALSE

I The first entry in the product Ax is a sum of products. TRUE

I If the columns of an m× n matrix span Rm, then the equationAx = b is consistent for each b in Rm TRUE

I If A is an m × n matrix and if the equation Ax = b isinconsistent for some b in Rm, then A cannot have a pivotposition in every row. TRUE

Linear Algebra, David Lay Week Two True or False

Section 1.4 24

I Every matrix equation Ax = b corresponds to a vectorequation with the same solution set. TRUE

I Any linear combination of vectors can always be written in theform Ax for a suitable matrix A and vector x . TRUE

I The solution set of the linear system whose augmented matrixis [a1 a2 a3 b] is the same as the solution set of Ax = b, ifA = [a1 a2 a3] TRUE

I If the equation Ax = b is inconsistent, then b is not in the setspanned by the columns of A. TRUE

I If the augmented matrix [A b] has a pivot position in everyrow, then the equation Ax = b is inconsistent. FALSE

I If A is an m × n matrix whose columns do not span Rm, thenthe equation Ax = b is inconsistent for some b in Rm. TRUE

Linear Algebra, David Lay Week Two True or False