Linear Algebra Final Exam

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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.


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