# Linear Algebra True False

true
every elementary row operation is reversible
false
a 5×6 matrix has six rows
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Linear Algebra True False
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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 , 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

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