FALSE – Every matrix is row equivalent to a unique matrix in ROW echelon form.
Every matrix is row equivalent to a unique matrix in echelon form.
FALSE – There could be no solution
Any system of n linear equations in n variables has at most n solutions
TRUE
If a system of linear equations has two different solutions, it must have infinitely many solutions
FALSE – It could have no solution
If a system of linear equations has no free variables, then it has a unique solution
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, then so does the system Ax=0
FALSE – It must be consistent for ALL b
If A is an m x n matrix and the equation Ax=b is consistent for some b, then the columns of A span Rm
FALSE – It can be in RREF, but still be inconsistent [0 0 0 | 5]
If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax=b is consistent.
TRUE
If matrices A and B are row equivalent, they have the same reduced echelon form.
FALSE – There is always a trivial solution
The equation Ax=0 has the trivial solution if and only if there are no free variables.
TRUE
If A is an m x n matrix and the equation Ax=b is consistent for every b in Rm, then A has m pivot columns.
TRUE
If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.
FALSE – There could be a column of zeros meaning there is a free variable and infinite solutions
If an m x n matrix A has a pivot position in every row. then the equation Ax has a unique solution for each b in Rm.
TRUE
If 3×3 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 m x n matrix, if the equation Ax=b has at lease 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 m x n matrices and if the columns of A span Rm, then so do the columns of B.
FALSE – Multiple does not mean linear combination. If there is a linear combination the set is dependent.
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.
TRUE
If {u,v,w} is linearly independent, then u, v, and w are not in R2
FALSE – There would not be a pivot position in every row
In some cases, it is possible for four vectors to span R5
TRUE
If u and v are in Rm, then -u is in Span{u,v}.
FALSE – It will still be in R2 if w is not a linear combination of u and v if u and v are on the same line.
If u, v, and w are nonzero vectors in R2, then w is a linear combination of v and u.
FALSE – It doesn’t say the vectors are nonzero
If w is a linear combination of u and v in Rn, then u is a linear combination of v and w.
FALSE – One could be the zero vector
Suppose that v1, v2, and v3 are in R5, 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
TRUE
If A is a 6×5 matrix, the linear transformation x –> Ax cannot map R5 ONTO R6
FALSE – There may not be a pivot in every column.
If A is an m x n matrix with m pivot columns, then the linear transformation x –> Ax is a one-to-one mapping
TRUE
Every elementary row operation is reversible.
FALSE – It has 5 rows and 6 columns.
A 5×6 matrix has six rows.
FALSE – The solution set is a collection of lists.
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…x2 respectively.
TRUE
Two fundamental questions about a linear system involve existence and uniqueness.
TRUE
Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
FALSE – Row equivalence means one matrix can be transformed into the other by elementary row operations.
Two matrices are row equivalent if they have the same number of rows.
FALSE – An inconsistent system has no solution.
An inconsistent system has more than one solution.
TRUE
Two linear systems are equivalent if they have the same solution set.
FALSE – A matrix may be reduced to more than one matrix in echelon form.
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 – It can be applied to all matrices.
The row reduction algorithm applies only to augmented matrices for a linear system.
TRUE
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
TRUE
Finding a parametric description of the solution set of a linear system is the same as solving the system.
FALSE – To be inconsistent the row would be [0 0 0 0 5]
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 – The reduced echelon form is unique
The echelon form of a matrix is unique.
FALSE
The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
TRUE
Reducing a matrix to echelon form is called the forward phase of the row reduction process.
FALSE – It may have no solution.
Whenever a system has free variables, the solution set contains many solutions.
TRUE
A general solution of a system is an explicit description of all solutions of the system.
TRUE
An example of a linear combination of vectors v1 and v2 is the vector .5v1
FALSE – It could be a line
The set Span{u,v} is always visualized as a plane through the origin.
TRUE
Any list of five real numbers is a vector in R5
TRUE
The vector u results when a vector u-v is added to the vector v.
FALSE – If they are it is just a point at the origin.
The weights c1,…cp in a linear combination c1v1+…+vpcp connot all be zero.
TRUE
When u and v are nonzero vectors, Span {u,v} contains the line through u and the origin.
TRUE
Asking whether the linear system corresponding to an augemented matrix [a1 a2 a3 b] has a soliton amounts to asking whether b is Span {a1,a2,a3}.
FALSE – It is a matrix equation.
The equation Ax=b is referred to as a vector equation.
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.
FALSE – If the pivot is in column b it will be inconsistent.
The equation Ax=b is consistent if the augmented matrix [A b] has a pivot position in every row.
TRUE
The first entry in the product Ax is a sum of products.
TRUE
If the columns of an m x n matrix A span Rm, then the equation Ax=b is consistent for each b in Rm.
TRUE
If A is an m x 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.
TRUE
Every matrix equation Ax=b corresponds to a vector equation with the same solution set.
TRUE
Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.
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].
TRUE
If the equation Ax=b is inconsistent, then b is not in the set spanned by the columns of A.
FALSE – When the matrix has more columns than rows.
If the augmented matrix [A b] has a pivot position in every row, then the equation Ax=b is inconsistent.
TRUE
If A is an m x n matrix whose columns do not span Rm, then the equation Ax=b is inconsistent for some b in Rm.
FALSE – It could have one entry be a 0.
If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.
TRUE
The equation x=x2u+x3v, with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin.
TRUE
The equation Ax=b is homogeneous if the zero vector is a solution.
TRUE
The effect of adding p to a vector is to move the vector in a direction parallel to p.
FALSE – They are not the same
The solution set of Ax=b is obtained by translating the solution set of Ax=0.
TRUE
A homogeneous equation is always consistent.
FALSE – It is just one solution – the trivial solution.
The equation Ax=0 gives an explicit description of its solution set.
FALSE – It ALWAYS has the trivial solution.
The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable
TRUE
The equations x=p+tv describes a line through v parallel to p.
FALSE -vh can be any solution.
The solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the equation Ax=0.
FALSE – They would be linearly dependent
The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution.
FALSE – One could be the zero vector
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
TRUE
The columns of any 4×5 matrix are linearly dependent.
TRUE
If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span {x,y}
TRUE
Two vectors are linearly dependent if and only if they lie on a line through the origin.
FALSE
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly dependent.
TRUE
If x and y are linearly independent, and if z is in Span {x,y}, then {x,y,z} is linearly dependent.
FALSE
If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector
TRUE
A linear transformation is a special type of function
FALSE – domain is R5
If A is a 3×5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is R3
FALSE – Range = T(x)
If A is an m x n matrix, then the range of the transformation x –> Ax is Rm
FALSE
Every linear transformation is a matrix transformation
TRUE
Every matrix transformation is a linear transformation
FALSE – The range
The codomain of the transformation x –> Ax is the set of all linear combinations of A.
FALSE – existence
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?”
TRUE
A linear transformation preserves the operations of vector addition and scalar multiplication
TRUE
A linear transformation T:Rn–>Rm is completely determined by its effect on the columns of the n x n identity matrix.
TRUE
If T:R2–>R2 rotates vectors about the origin through and angle then T is a linear transformation.
FALSE
When two linear transformations are performed one after another, the combined effect mat not always be a linear transformation.
FALSE
A mapping T:Rn–>Rm is onto Rm if every vector x in Rn maps onto some vector in Rm
FALSE
If A is a 3×2 matrix, then the transformation x –> Ax cannot be 1-1
