# Linear algebra test true and false

a homogeneous equation is always consistent

TRUE Ax=b always has “trivial” solution (where all the variables are 0)

the homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable

FALSE Ax=0 always has the trivial solution

the equation Ax=0 gives an explicit description of its solution set

FALSE the original equation is the implicit description and the explicit description is the equation solved for the span

the equation x=p + tv describes a line through v parallel to p

FALSE describes a line through p parallel to v

a homogeneous system of equations can be inconsistent

FALSE always has to have one solution x=0

if x is a nontrivial solution of Ax=0, then every entry in x is nonzero

FALSE, can have some 0 entries where x=0 but not all

the equation Ax=b is a homogeneous if the zero vector is a solution

TRUE the zero vector is always a solution to homogeneous systems

the columns of A are linearly independent if the equation Ax=0 has the trivial solution

FALSE a homogeneous system always has the trivial solution

if S is a linearly dependent set, then each vector is a linear combination of the other vectors S

FALSE not every vector in a linearly dependent set is a combination of the preceding vectors

the columns of any 4×5 matrix are linearly dependent

TRUE is there are more columns than rows then its linearly dependent

if x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span {x,y}

TRUE {x, y, z} will be linearly dependent if and only if w is on the plane spanned by u and v

if u and v are linearly independent, and if w is in the span {u, v}, then {u, v, w} is linearly dependent

TRUE S={v1…vp} of 2 or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others

if three vectors in R3 lie in the same plane in R3, then they are linearly dependent

TRUE the third vector must be a multiple of the first or the second vector (who can’t be multiples of each other which means they are linearly independent)

if a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent

FALSE set consisting of

v1= v2=

is linearly dependent, the theorem p>n does not apply when p

if a set in Rn is linearly dependent, then the set contains more than n vectors

FALSE can have the same number of vectors as n

v1= v2=

v2 is a multiple of v1 so {v1, v2} is linearly dependent and has the same number of vectors

v1= v2=

v2 is a multiple of v1 so {v1, v2} is linearly dependent and has the same number of vectors

if v1…v4 are in R4 and v3=2v1+v2 then {v1, v2, v3, v4} is linearly dependent

TRUE S={v1…vp} of 2 or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others

if v1 and v2 are in R4 and v2 is not a scalar multiple of v1, then {v1,v2} is linearly dependent

FALSE the vector v1 could be the zero vector

if v1,…v5 are in R5 and v3=0 then {v1, v2, v3, v4, v5} is linearly dependent

TRUE if a set S={v1…vp} in Rn contains the zero vector, then the set is linearly dependent

if v1, v2, v3 are in R3 and v3 is not a linear combination of v1, v2 then {v1, v2, v3} is linearly independent

FALSE v1 and v2 can be multiples of each other making the system linearly dependent

if v1…v4 is in R4 and {v1,v2,v3} is linearly dependent, then {v1, v2, v3, v4} is also linearly dependent

TRUE a linear dependence relation among v1, v2, v3 may be extended to linear dependence relation among v1, v2, v3, v4 by placing a zero weight on v4

if {v1…v4} is a linearly independent set of vectors in R4 then {v1, v2, v3} is also linearly independent

TRUE if the equation x1v1+x2v2+x3v3+0 x v4=0 had a normal solution with at least one of the other three vectors being nonzero, then so would the equation x1v1+x2v2+x3v3+0 x v4=0. but that cannot happen because {v1, v2, v3, v4} is linearly independent. so {v1, v2, v3} must be linearly independent.

a linear transformation is a special type of function

TRUE a linear transformation is a function with certain properties

if A is a 3 x 5 matrix and T is a transformation defined by T(x)=Ax then the domain of T is R3

FALSE the domain is R5, the domain of T is the number of columns

if A is an m x n matrix, then the range of the transformation x |–> Ax is Rm

FALSE the range is the set of all linear combinations of the columns of A, because each image T(x) is of the form Ax

every linear transformation is a matrix transformation

FALSE every matrix transformation is a linear transformation, but not the reverse

a transformation T is linear if and only if 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 linear transformations preserve the operations of vector addition and scalar multiplication

the range of the transformation x |–> Ax is the set of all linear combinations of the columns of A

TRUE the range is the set of all linear combinations of the columns of A, because each image T(x) is of the form Ax

every matrix transformation is a linear transformation

TRUE every matrix transformation is a linear transformation, but not the reverse

if T: Rn –> Rm is a linear transformation and if c is in Rm, then a uniqueness question is “Is c in the range of T?”

FALSE this is an existence question, another way of asking if Ax=C is consistent

A linear transformation preserves the operations of vector addition and scalar multiplication

TRUE linear transformations preserve the operations of vector addition and scalar multiplication

a linear transformation T: Rn –> Rm always maps the origin of Rn to the origin of Rm

TRUE If T is a linear transformation then T(0)=0, and T(cu+dv)=cT(u)+dT(v)