Linear Algebra Final Exam
A linear system can have exactly two solutions
False
A set S={v1,v2,...,vk} k>=2, is linearly independent if and only if at least one of the vectors vj can be written as a linear combination of the other vectors
False
A set of vectors S={v1,v2,...,vk} is a vector space is called linearly dependent with the vector equation has only the trivial solution
False
A system of three linear equations in two variables is always inconsistent
False
A system of two linear equations in three variables is always consistent
False
All linear transformations T have a unique inverse T^-1
False
Any linear transformation of the form f(x)=ax+b is a linear transform from R into R
False
Ax=0 has only the trivial solution if and only if Ax=b has a unique solution for every nx1 column matrix b
False
If A is a square matrix, then the system of linear equations Ax=b has a unique solution
False
If A is an mxn matrix of rank r, then the dimension fo the solution space of Ax=0 is m-r
False
If E is an elementary matrix, then 2E is an elementary matrix
False
If P is the transition matrix from a basis B to B', then the equation P[x]B'=[x]B represents the change of basis from B to B'
False
If U,V,W are vector spaces such that W is a subspace of V and U is subspace of V, then W=U
False
If a linear system is consistent, then it has infinitely many solutions
False
If an mxn matrix B can be obtained from elementary row operations on an mxn matrix A. then the column space of B is equal to the column space of A
False
If dim(V)=n, then any set of n-1 vectors in V must be linearly independent
False
If dim(V)=n, then there exists a set of n-1 vectors in V that will span V
False
If the matrices A,b,C satisfy AB=AC, then B=C
False
If the row echelon form of the augmented matrix of a system of linear equations contains the row [1,0,0,0,0] then the original system is inconsistent
False
In general, the compositions T2oT1 and T1oT2 have the same standard matrix A
False
Matrix multiplication is commutative
False
The coordinate matrix of p=5x^2+x-3 relative to the standard basis for P2 is [p]S=[5 1 -3]^T
False
The dimension of a linear transformation T from a vector space V into a vector space W is called the rank of T
False
The function f(x)=cosx is a linear transformation for R into R
False
The function g(x)=x^3 is a linear transformation from R into R
False
The inverse of the product of two metrics is the product of their inverses; that is, (AB)^-1=A^-1B^-1
False
The nullspace of a matrix A is also called the solution space of A
False
The range of linear transformation form a vector space V into a vector space W is a subspace of V
False
The scalar λ is an eigen value of an nxn matrix A when there exists a vector x such that Ax=λx
False
The set of all first degree polynomials with standard operations is a vector space
False
The set of all integers with the standard operations is a vector space
False
The set of all ordered triples (x,y,z) of real numbers, where y>=0, with the standard operations is a vector space
False
The set of all vectors mapped from a vector space V into another vector space W by a linear transformation T is the kernel of T
False
The standard basis for R^n will always make the coordinate matrix for the linear transformation T the simplest matrix possible
False
The system of linear equations Ax=b is inconsistent if and only if b is in the column space of A
False
The transpose of the product of two metrics equals the product of their transposes that is (AB)^T=A^TB^T
False
The vector -v is called the additive identity of the vector v
False
The zero matrix is an elementary matrix
False
The zero vector 0 in R^n is defined as the additive inverse of a vector
False
To perform the change of basis from a nonstandard basis B' to the standard basis B, the transition matrix P is simply B'
False
Two matrices that represent the same linear transformation T:V-->V with respect to different bases are not necessarily similar
False
A homogenous system of four linear equations in four variables is always consistent
True
A linear transformation T from V into W is one to one when the preimage of every w in the range consists of a single vector v
True
A square matrix is nonsingular when it can be written as the product of elementary matrices
True
A system of one linear equation in two variables is always consistent
True
A vector space consists of four entries: a set of vectors, a set of scalars, and two operations
True
Every matrix A has an additive inverse
True
Every matrix is row-equivalent to a matrix in row-echelon form
True
Every vector space V contains at least one subspace that is the zero subspace
True
For any 4x1 matrix X, the coordinate matrix [X]S relative to the standard basis for M4,1 is equal to X itself
True
For any matrix C, the matrix CC^T is symmetric
True
For polynomials, the differential operator Dx is a linear transformation from Pn into Pn-1
True
If A is an nxn matrix with an eigenvalue λ, then the set of all eigenvectors of λ is a subspace of R^n
True
If V and W are both subspaces of a vector space U, then the intersection of V and W is also a subspace
True
If a subset S spans a vector space V, then every vector in V can be written as a linear combination of the vectors in S.
True
If any matrix A is row-equivalent to an mxn matrix B, then the row space of A is equivalent to the row space of B.
True
If dim(V)=n, then any set of n+1 vectors in V must be linearly dependent
True
If dim(V)=n, then there exists a set of n+1 vectors in V that will span V
True
If the matrices A,B,C satisfy BA=CA and A is invertible, then B=C
True
Matrix addition is commutative
True
Matrix multiplication is associative
True
The column space of a matrix A is equal to the row space of A^T
True
The composition T of linear transformations T1 and T2, represented by T(v)=T2(T1(v)) is defined when the range of T1 lies with the domain of T2
True
The identity matrix is an elementary matrix
True
The inverse of an elementary matrix is an elementary matrix
True
The inverse of the inverse of a nonsingular matrix A is equal to itself
True
The null space of a matrix A is the solution space of the homogenous system Ax=0
True
The set S={(1,0,0,0),(0,-1,0,0),(0,0,1,0),(0,0,0,1)} spans R^4
True
The set of all pairs of real numbers of the form (0,y) with the standard operations on R^2 is a vector space
True
The system Ax=b is consistent if and only if b can be expressed as a linear combination of the columns of A, where the coefficients of the linear combination are a solution of the system
True
The transpose of the same of two matrices equals the sum of their transposes
True
The vector spaces R^2 and P1 are isomorphic to each other
True
The vector spaces R^3 and M3,1 are isomorphic to eachother
True
To subtract two vectors in R^n, subtract their corresponding components
True
Two systems of linear equations are equivalent when they have the same set
True
Two vectors in R^n are equals if and only if their corresponding components are equal
True
You can have λ=0
True
A homogenous system of four linear equations in six variables has infinitely many solutions
True. More variables than equations so infinite.