Linear Algebra Exam 1
(ABC)^T = C^T*A^T*B^T
false
if A is a 4x3 matrix, then x->Ax maps R3 onto R4.
false
if a set in R^n is linearly dependent, the set contains more than n vectors.
false
the weights c1...cp in a lin. combination cannot all be zero
false
The solution set of Ax=b is the set of all vectors of the form w=p+v, where v is any solution of the equation Ax=0.
true
the effect of adding p to a vector is to move the vector in the direction parallel to p.
true
the ref of a matrix is unique
true
Two fundamental questions about a linear system involve existence and uniqueness.
True
The equation Ax=0 gives an explicit description of its solution set.
false
The homogeneous equation Ax=0 has the trivial solution iff the equation has at least one free variable.
false
The row reduction algorithm applies only to augmented matrices for a linear system.
false
The set span{u...v} is always visualized as a plane through the origin
false
The transpose of a product of matrices equals the product of their transposes in the same order.
false
When 2 linear transformations are performed one after another, the combined effect may not always be a linear transformation.
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
if every column of an augmented matrix contains a pivot, the system is consistent.
false
if one row in echelon form of an augmented matrix is [0 0 0 5 0], then the linear system is inconsistent.
false
whenever a solution has free variables, the solution set contains many solutions.
false
If a set contains fewer vectors than there are entries in the vectors, the set is linearly dependent.
false (conditional (?))
If A is an mxn matrix, the range x->Ax is Rm.
false - Rm is the codomain. The range is where we actually land.
Every trans. is a matrix trans.
false - every matrix tran is a linear tran.
In some cases, a matrix may be row reduced to more than one matrix in REF.
False
If A and B are nxn and invertible, then A^-1 * B^-1 is the inverse of AB.
False - B^-1*A^-1
A mapping T: Rn->Rm is one to one if each vector in Rn maps onto a unique vector in Rm.
false - mapping is one to one if each vector in Rm is mapped to from a unique vector in Rn
If A and B are 2x2 matrices with columns a1, a2, and b1, b2, respectively, then AB = [a1b1, a2b2]
false - matrix multiplication is row by column
If A & B are 3x3 and B = [b1 b2 b3], then AB = [Ab1+Ab2+Ab3]
false - no plus signs in the answer
if A is nxn, Ax=b has at least one solution for each b in Rn.
false - not enough information
if A is a 3x2 matrix, the nthe transformation x->Ax is one to one.
false - since it maps from R2 to R3 and 2<3 it can be one to one but not onto.
If T: Rn->Rm is a linear trans. and if c is in Rm, then a uniqueness question is "Is c in the range of T?"
false - that is an existence question.
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
false, only one has to be
If A is invertible, then row operations that reduce A to I_n also reduce A^-1 to I_n.
false, they reduce I_n to A^-1
The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution.
false, unless ONLY the trivial solution
If A is a 3x5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is R3.
false. domain is R5.
A homogeneous equation is always consistent.
true
A linear trans. is a special kind of function.
true
A linear trans. preserves the operations of vector addition and scalar multiplication.
true
A linear transformation is completely determined by the effect on the columns of the nxn identity matrix.
true
A^T+B^T = (A+B)^T
true
Ax=b is homogeneous if the zero vector is a solution.
true
Each column of AB is a linear combination of the columns of B using weights from the corresponding columns of A.
true
Every lin. trans. from Rn to Rm is a matrix transformation.
true
Every matrix trans. is a linear trans.
true
If A can be row reduced to the identity matrix, then A is invertible.
true
If A is an nxn and Ax=e_j is consistent for every j from 1 to n, then A is invertible.
true
If A is an nxn matrix, then (A^2)^T = (A^T)^2
true
If A is invertible, then the inverse of A^-1 is A itself.
true
If A^T is not invertible, then A is not invertible.
true
If Ax=b is consistent, the solution set of Ax=b is obtained by translating the solution set of Ax=0.
true
If Ax=b is consistent, then b is in the set spanned by the columns of A.
true
If T: R2->R2 rotates vectors about the origin through an angle phi, then T is a linear transformation.
true
If the columns of A are linearly independent, then the columns of A span Rn.
true
If the columns of A span Rn, the columns are linearly independent.
true
If the columns of an mxn matrix A span R^m, then the equation Ax=b is consistent for each b in R^m.
true
If the equation Ax=0 has a nontrivial solution, then A is row equivalent to the nxn identity matrix.
true
If there is an nxn matrix D such that AD=I, then DA=I.
true
If u and v are lin. independent, and if w is in span[u,v], then {u,v,w} is linearly dependent.
true
In order for a matrix B to be the inverse of A, the equations AB=I and BA=I must both be true.
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 under t.
true
The first row of AB is the first row of A multiplied on the right by B
true
The range of the transformation x->Ax is the set of all linear combinations of the columns of A.
true
The standard matrix of a horizontal shear trans. from R2->R2 has the form [(a 0 / 0 d)] where a and d are +-1.
true
a general solution of a system is an explicit description of all solutions of a system.
true
any list of 5 real numbers is a vector in R5.
true
each elementary matrix is invertible
true
finding a parametric description of a system is the same as solving a system.
true
if Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
true
if a linear system [a1 a2 a3 b] has a solution is equivilent to saying b is in span{a1,a2,a3}
true
if there is a b in Rn such that Ax=b is consistent, then the solution is unique.
true
if x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span{x,y}
true
the transpose of a sum of matrices equals the sum of the transposes.
true
AB + AC = A(B+C)
true - matrix multiplication distributes over addition
The columns of any 4x5 matrix are linearly dependent.
true, more vectors than entries.
A 5x6 matrix has 6 rows
True
A basic variable in a linear system is a variable that corresponds to a pivot column in a coefficient matrix.
True
A lin. trans. T:Rn->Rm always maps the origin of Rn to the origin of Rm.
True
Every Elementary row operation is reversible
True
A mapping T: Rn->Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.
false
If Ax=b has at least one solution for each b in Rn, then x->Ax is not one-to-one.
false
If the lin. trans. x->Ax maps Rn into Rn, then the row reduced echelon form of A is I.
false
If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.
false
