Linear Algebra Exam 2
The transpose of a product of matrices equals the product of their transpose in the same order
False
Given vectors v1...vp in Rn the set of all linear combinations of these vectors is a subspace of Rn
True
If A = (ab;cd) and ad=bc then A is not invertible
True
If A can be row reduction to the identity matric then A must be invertible
True
If A is a 3x2 matrix then the transformation x-> Ax cannot map R2 onto R3
True
If A is an invertible nxn matrix then the equation Ax = b is consistent for each b in IR
True
If A is invertible then the inverse of A-1 is A itself
True
If B = {v1...vp} is a basis for a subspace H and if x = c1v1+...cpvp then c1...cp are the coordinated of x relative to the basis B
True
If B = {v1...vp} is a basis for a substance H of Rn then the correspondence x->[x]B makes H look and act the same as Rn
True
If B is a basis for a subspace H then each vector in H can be written in only one way as a linear combination of the vectors in B
True
If H is p-dimensional subspace of Rn then linearly independent set of p vectors in H is a basis for H
True
If a set of P vectors spans a p-dimensional subspace H of Rn then the vectors form a basis for H
True
If the Equation Ax = b has at least one solution for each b in IRn then the solution is unique for each b
True
If the columns of A are linearly independent then the columns of A span IRn
True
If the columns of A span IRn then the columns are linearly independent
True
If the equation Ax = 0 has a nontrivial solution then A has fewer then n pivot positions
True
If the equation Ax = 0 has only trivial solutions then A is row equivalent to the nxn matrix
True
If there is a b in IRn such that the equation Ax = b is inconsistent then the transformation x-> Ax is not one to one
True
If there is an nxn matric D such that AD = I then there is also an nxn matrix C such that CA = I
True
In order for a matrix B to be the inverse of A both equation AB=I and BA=I must be true
True
Row operations do not affect linear dependence relations among the columns of a matrix
True
The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the nxn identity matrix
True
The second row of AB is the second row of A multiplied on the right by B
True
AB+AC = A(B+C)
True
If AT is not invertible then A is not invertible
True
If v1...vp are in Rn then span {v1...vp} is the same as the column space of the matrix (v1...vp)
True
linear transformation is a special type of function
True
The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis the vertical axis or the origin had the form (a0;0d) where a and d are + or - 1
True
The superposition principle is a physical description of linear transformation
True
The transpose of a sum of matrices equals the sum of their transpose
True
if T: R2-> R2 rotates vectors about the origin through an angle then T is a linear transformation
True
the columns of an invertible nxn matrix form a basis for Rn
True
the dimension of ColA is the number of pivot columns of A
True
the dimension of the column space of A is rank A
True
the dimensions of ColA and NulA add up to the number of column A
True
the null space of an mxn matrixx is a subspace of Rn
True
the set of all solutions of a system of m homogeneous equations in n unknown is a subspace of Rm
True
(AB)C = (AC)B
False
(AB)T = ATBT
False
A mapping T: Rn->Rm is one to one if each vector in Rn maps onto a unique vector in Rm
False
A mapping T: Rn->Rm is onto Rm if every vector x in Rn maps onto some vector in Rm
False
A product of invertible nxn matrices is invertible and the inverse of the product is the product of their inverses in the same order
False
A subspace of Rn is any set H such that (i) the zero vector is in H(ii) u,v and u+v are in H and (iii) c is a scalar
False
Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
False
Each line in Rn is one-dimensional subspace of Rn
False
If A = (a b; cd) and ab-cd does not equal zero then A is invertible
False
If A and B are 2x2 with columns a1, a2 and b1, b2 respectively, then AB = [a1b1 a2b2]
False
If A and B are 3x3 matrices and B=[b1 b2 b3], then AB= [Ab1 + Ab2 + Ab3]
False
If A and B are nxn and invertible then A-1B-1 is the inverse of AB
False
If A is a 3x5 matrix and T is a transformation defined by T(x)=Ax then the domain of T is R3
False
If A is a 3xn matrix then the transformation x->Ax cannot be one to one
False
If A is an mxn matric then the range of the transformation x-> Ax is Rm
False
If A is invertible then elementary row operations that reduce A to identity in also reduce A-1 to In
False
If B is an echelon form of a matrix A then the pivot columns of B form a basis ColA
False
If T: Rn -> Rm is a linear transformation and if c is in Rm then the uniqueness question is c in the range of T
False
If the linear transformation x -> Ax maps IRn into IRn then A has n pivot positions
False
Not every linear transformation form Rn to Rm is a matrix transformation
False
The codomain of the transformation x-> Ax is the set of all linear combination of the columns of A
False
The column space of a matrix A is the set of solutions of Ax = b
False
When 2 linear transformations are preformed one after another the combined effect may not always be linear transformation
False
the dimension of NulA is the number of variables in the equation Ax=0
False
A subset H of Rn is a subspace if the zero vectors is in H
False
A linear transformation preserves the operations of vector addition and scalar multiplication
True
Every linear transformation is a matrix transformation
False
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 and c2
True
AT + BT = (A+B)T
True
Each elementary matrix is invertible
True
Every matrix transformation is a linear transformation
True
