True
If a system of linear equations has at least two solutions, it has infinitely many solutions
True
If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets
True
If a system Ax=b has more than one solution, so does Ax=0
True
If matrices A and B are row equivalent, then they have the same rref
True
If A is an mxn matrix and the equation Ax=b is consistent for every b in R^m, then A has m pivot columns
True
If an n xn matrix A has n pivot positions, then the reduced echelon form of A is the nxn identity matrix
True
If 3 x3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations
True
If A is an mxn matrix, if the equation Ax=b has at least two different solutions, and if the equation Ax=c is consistent, then the equation Ax=c has many solutions
True
If A and B are row equivalent mxn matrices and if the columns of A span R^m, then so do the columns of B
True
If {u, v, w} is linearly independent, then u, v, and w are not in R^2 .
True
If u and v are in R^m, then u is in Span{u, v}
True
Suppose that v1, v2, and v3 are in R^5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1, v2, v3} is linearly independent.
True
A linear transformation is a function.
False
Every matrix is row equivalent to a unique matrix in echelon form
False
Any system on n linear equations in n variables has at most n solutions
False
If a system of linear equations has no free variables, then the solution is unique
False
If A is an m x n matrix and the equation Ax = b is consistent for some b, then the columns of A span R^m.
False
If an augmented matrix [A b] can be transformed by elementary operations into reduced echelon form, then Ax=b is consistent
False
The equation Ax=0 has the trivial solution iff it has no free variables
False
If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a unique solution for each b in R^m
False
If none of the vectors in the set S{v1;v2;v3} in R3 is a multiple of one of the other vectors, then S is linearly independent
False
In some cases, it is possible for four vectors to span R^5
False
If u, v, and w are nonzero vectors in R^2 , then w is a linear combination of u and v
False
If w is a linear combination of u and v in R^n,then u is a linear combination of v and w
False
If A is a 6X5 matrix, the linear transformation x↦Ax cannot map R^5 onto R^6
False
If A is an mxn matrix with m pivot columns, then the linear transformation x↦Ax is a one-to-one mapping
False
A 5×6 matrix has 6 rows.
True
The solution set of a 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.
True
A consistent system of linear equations has one or more solutions
True
Two fundamental questions about a linear system involve existence and uniqueness
True
Elementary row operations on an augmented matrix never change the solutions set of the associated linear system
False
Two matrices are row equivalent if they have the same number of rows.
True
Every elementary row operation is reversible
False
Two equivalent linear systems can have different solution sets
False
The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process
True
Whenever a system has free variables, the solution set contains many solutions
True
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
True
The reduced echelon form of a matrix is unique.
False
The row reduction algorithm applies only to augmented matrices for a linear system.
True
A general solution of a system is an explicit description of all solutions of the system
False
If every column of an augmented matrix contains a pivot, then the corresponding system is consistent
False
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations
False
If one row in an echelon form of an augmented matrix is [0 0 0 4 0] then the associated linear system is inconsistent
True
Finding a parametric description of the solution set of a linear system is the same as solving that system.
True
The solution set of a linear system whose augmented matrix is [a1 a2 a3 b]is the same as the solution set of the equation x1a1+x2a2+x3a3=b
True
An example of a linear combination of vectors v1 and v2 is the vector .5v1
True
Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span{a1, a2, a3}
False
The weights c1, … cp in a linear combination c1v1+…+cpvp cannot all be zero
True
Any list of five real numbers is a vector in ℝ5
False
The set Span{u, v} is always visualized as a plane through the origin.
False
The vector v results when a vector u-v is added to the vector v
False
When u and v are nonzero vectors, Span{u, v} contains only the line through u and the origin and the line through v and the origin.
False
The points in the plane corresponding to [3 -4] and [4 -3] lie on a line through the origin
True
If AA is an m×n matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row.
False
The equation Ax=b is consistent if the augmented matrix [A b] has a pivot position in every row
True
Every matrix equation Ax=b corresponds to a vector equation with the same solution set
True
If the columns of an m×n matrix span ℝm, then the equation Ax=b is consistent for each b in Rm
True
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.
True
Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x
True
If the equation Ax=b is consistent, then b is in the set spanned by the columns of A
False
If the coefficient matrix A has a pivot position in every row, then the equation Ax=b is inconsistent.
False
The equation Ax=b is referred to as a vector equation
False
If A is an m×n matrix whose columns do not span ℝm then the equation Ax=b is consistent for every b in ℝm.
True
The first entry in the product Ax is a sum of products
True
The solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax=b, if A = [a1 a2 a3]
False
The equation Ax=0 gives an explicit description of its solution set
True
A homogeneous equation is always consistent
False
The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable
True
The effect of adding p to a vector is to move the vector in a direction parallel to p
False
If x is a nontrivial solution of Ax=0, then every entry in x is nonzero
False
A homogeneous system of equations can be inconsistent
True
The equation Ax=b is homogeneous if the zero vector is a solution.
False
The equation x=p+tv describes a line through v parallel to p
True
If Ax=b is consistent, then the solution set of Ax=b is obtained by translating the solution set of Ax=0
True
The solution set of Ax=b is the set of all vectors of the form w = p + vh, where vh is any soluotion of the equation Ax=0 and p is one particular solution of Ax=b
False
If a set contains fewer vector than there are entries in the vectors, then the set is linearly independen
True
If three vectors in R3 lie in the same plane in R3, then they are linearly dependent.
True
The columns of any 4×5 matrix are linearly dependent
False
If a set in Rn is linearly dependent, then the set contains more than n vectors.
True
If x and y are linearly independent, and if {x ,y ,z} is linearly dependent, then z is in Span{x,y}
False
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
True
The columns of a matrix A are linearly independent if the equation Ax=0 has only the trivial solution
True
If u and v are linearly independent, and if u is in Span{u,v} then {u,v,w} is linearly dependent.
False
If v1, v2, v3 are in R3 and v3 is not a linear combination of v1and v2, then {v1, v2, v3} is linearly independent
True
Every matrix transformation is a linear transformation
True
A linear transformation preserves the operations of vector addition and scalar multiplication
True
A linear transformation is a special type of function.
False
If T:Rn –> Rm is a linear transformation and if c is in Rm, then a uniqueness question is: Is c in the range of T?
False
Every linear transformation is a matrix transformation.
True
A transformation TT is linear if and only if T(c1v1+c2v2)=c1T(v1)+c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.
True
A linear transformation ℝn→ℝm always maps the origin of ℝn to the origin of ℝm.
False
If A is a 3×5 matrix and T is a transformation defined by T(x )=Ax, then the domain of T is ℝ3
True
The range of a transformation x–>Ax is the set of all linear combinations of the columns of A.
False
If AA is an m×nm×n matrix, then the range of the transformation x→Ax→ is ℝm
