true

every elementary row operation is reversible

false

a 5×6 matrix has six rows

false

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

two fundamental questions about a linear system involve existence and uniqueness

false

two matrices are row equivalent if they have the same number of rows

true

elementary row operations on an augumented matrix never change the solution set of the associated linear system

false

two equivalent linear systems can have different solution sets

true

a consistent system of linear equations has one or more solutions

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

the row reduction algorithm applies only to augmented matrices for a linear system

true

a basic variable in a linear system is a varaible 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

if one row in an echelon form of an augmented matrix is [00050], then the associated linear system is inconsistent

true

the reduced echelonf form of a matrix is unique

false

if every column of an augmented matrix contains a pivot then the corresponding system is consistent

false

the pivot positions in a matrix depend on whether row interchanges are used in the row reduction process

true

a general solution of a system is an explicit description of all solutions of the system

false

whenever a system has free variables the solution set contains many solutions

false

Another notation for the vector <-4, 3> is [-4 3].

false

the points in the plane corresponding <-2, 5> and <-5, 2> lie on a line through the origin

true

an example of a linear combination of vectors v1 and v2 is the vector 1/2*v1

true

the solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2+x3a3=b

false

the set span{uv} is always visualized as a plane through the origin

false

when u and v are non-zero vectors, span{uv} contains only the line through u and the origin and hte line through v and the origin

true

any list of 5 real numbers is a vector in R5

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 vector v results when a vector u-v is added to the vector v

false

the weights c1……cp in a linear combination c1v1+…+cpvp cannot all be 0

false

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

the equation ax=b is consistent if the augmented matrix [ab] 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 in 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

if the equation ax=b is consistent, then b is in the set spanned by the columns of A

true

any linear combination of vectors can always be written in the form ax for a suitable matrix a and vector x

false

if the coefficient matrix A has a pivot position in every row, then the equation ax=b is inconsistent

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

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

a homogenous equation is always consistent

false

the equation ax=0 gives an explicit description of a solution set

false

the homogenous solution ax=0 has the trivial solution iff the equation has at least one free variable

false

the equation x=p+tv describes a line through v parallel to p

true

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

a homogenous system of equations can be inconsistent

false

if x is a nontrivial solution of ax=0 then every entry in x is non-zero

true

the effect of adding p to a vector is to move the vector in a direction parallel to p

true

the equation ax=b is homogenous if the zero vector is a solution

true

if ax=b is consistent, then the solution set of ax=b is obtained by translating the solution set of ax=0

false

the columns of the matrix a are linearly independent if the equation ax=0 has the trivial solution

true

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 x 5 matrix are linearly dependent

true

if x and y are linearly independent, and if {xyz} is linearly independent, then z is in span{xy}

true

if u and v are linearly independent, and if w is in span {uv}, then {uvw} is linearly dependent

true

if 3 vectors in R3 lie in the same plane in R3, then they are linearly dependent

false

if a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent

false

if a set in Rn is linearly dependent, then the set contains more than n vectors

true

a linear transformation is a special type of function

false

if a is a 3×5 matrix and T is a transformation defined by Tx=Ax, then the domain of T is R3

true

If A is an m x n matrix then the range of the transformation X -> Ax is Rm

true

every linear transformation is a linear transformation

true

a transformation T is linear iff T(c1v1+c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 & c2`

true

the range of the transformation x -> Ax is the set of all linear combinations of the columns of A

true

every matrix transformation is a linear transformation

false

if T: Rn -> Rm is a linear transformation and if c is in Rm, then a uniqeness 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 always maps the origin of Rn to the origin of Rm

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 an angle theta, then T is a linear transformation

false

when 2 linear transformations are performed one after another the combined effect may 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 one-to-one

false

if a is a 4×3 matrix, then the transformations x->Ax maps R3 onto R4

true

every linear transformation from Rn -> Rm is a matrix transformation

true

the columns of the standard matrix of a linear transformation from Rn – > Rm are the images of the columns of the m x n identity matrix under T

false

a mapping T: Rn->Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm

true

the standard matrix of the horixzontal shear transformation from R2->R2 has the form , where a and d are +/- 1

false

If A and B are 2×2 matrices with columns a1,a2 and b1,b2 respectively, then ab = [a1b1 a2b2]

false

each column of AB is a linear combo of the columns of B using weights from the corresponding columns of A

true

AB + AC = A(B+C)

true

A^T + B^T = (A+B)^T

false

the transpose of a product of matrices = the product of their transposes in the same order

true

the first row of AB is the first row of A multiplied on the right by B

false

if A and B are 3×3 matrices and B =[b1, b2, b3], then AB = [Ab1 + Ab2 + Ab3]

true

if A is nxn, then (A^2)^T = (A^T)^2

false

(ABC)^T=C^TA^TB^T

true

the transpose of a sum of matrices = the sum of their transposes

true

in order for a matrix b to be the inverse of a, the equations ab=I and ba=I must both be true

false

if a and b are nxn and invertible then a^-1b^-1 is the inverse of ab

true

if a is an invertible nxn matrix then the equation ax=b is consistent for each b in Rn

true

each elementary matrix is invertible

false

if a is invertible then elementary row operations that reduce a to the identity In also reduce A^-1 to In

true

If A is invertible then the inverse of A^-1 is A

false

a product of invertible nxn matrices is invertible and hte inverse of the product is the product of their inverses in the same order

true

if a is nxn and ax=ej is consistent for every j belonging to {1,2…n} then a is invertible. Note: e1…en represent the columns of the identity matrix

false

if a can be row reduced to I, then A must be invertible

true

if the equation ax=0 has only the trivial solution then a is row equivalen tot the nxn identity matrix

true

if the columns of A span Rn then the coluumns are linearly independent

false

if a is nxn then the equation ax=b has at least one soltn. for each b in Rn

true

if the equation ax=e has a non-trivial soltn. then a has fewer than n pivot positions

true

if A^T is not invertible then A is not either

false

If there is an nxn matrix D such that AD=I then DA=I

true

if the linear transformation x->Ax maps Rn into Rn then the row reduced echelon form of A is I

true

if the columns of a are linearly independ. then the columns of A span Rn

false

if the equation ax=b has at least 1 soltn. for each b in Rn then the transformations x->Ax are not one-to-one

false

if there is a b in Rn such that the equation Ax=b is consistent then the slution is unique

true

A row replacement operation doesn’t effect the determinant of a matrix

true

the determinant of A is the product of the pivots in any echelon form U of A times -1^r where r is the number of row interchanges made during row reduction from A to U

false

if the columns of A are linearly depend. then detA=0

false

det(A+B) = detA + detB

true

if 2 row interchanges are made in succession then the new determinant = the new one

false

the detA is the product of the diagonal entries in A

true

if detA=0 then 2 rows/columns are the same or a row or a column = 0

false

det(A^T)=(-1)detA

false

A linear transformation T: R^5–>R^7 cannot be one to one

true

if a linear transformation T: R^4–>R^4 is one to one, it is also onto

false

If A and B are square matrices, then (AB)^T = A^TB^T

true

If A is a 2×2 matrix with detA = 0, the one column of A is a multiple of the other

true

If A^3=0, then detA =0

false

If A^3=0, then A =0

false

If A is a 3×3 matrix with detA=0, then one column of A is a multiple of the other