LINEAR ALGEBRA EXAM GUIDE
QUESTIONS WITH CORRECT ANSWERS
Consistent System - Answer-a linear system with at least one solution
inconsistent system - Answer-a linear system with no solution
linearly independent vectors - Answer-An indexed set {v1 , ... , vp} with the property that
the vector equation c1 v1 + c2 v2 + · · · + cp vp = 0 has only the trivial solution,
c1=···=cp=0.
linearly dependent vectors - Answer-An indexed set {v1, ... , vp} with the property that
there exist weights c1,•••,cp, not all zero, such that c1 v1 + · · · + cp vp = 0. That is, the
vector equation c1 v1 + c2 v2 + · · · + cp vp = 0 has a nontrivial solution.
Domain (of a transformation T: R"→R"') - Answer-The set of all vectors x for which T(x)
is defined (R"')
Codomain (of a transformation T: R"→R"') - Answer-The set Rm that contains the range
of T. In general, if T maps a vector space V into a vector space W, then W is called the
codomain of T (R")
Range (of a transformation T: R"→R"') - Answer-The set of all vectors (b) that live in the
codomain such that there is a vector (x) in Rm with T(x)=b
Linear Transformation - Answer-A rule T that assigns to each vector x in V a unique
vector T(x) in W, such that...
(i) T(u + v) = T(u) + T(v) for all u, v in V
(ii) T(cu) = cT(u) for all u in V and all scalars c.
Notation: T: V→W; also, x→Ax when T: R"→R"' and A is the standard matrix for T.
one-to-one transformation - Answer-A mapping T: R"→R"' such that each b in R"' is the
image of at most one x in R"
Onto Transformation - Answer-A mapping T: R"→R"' such that each b in R"' is the
image of at least one x in R"
Invertible Transformation - Answer-A linear transformation T: R"→R" such that there
exists a function S: R"→R" satisfying both T(S(x))=x and S(T(x))=x for all x in R"
Invertible (Nonsingular) Matrix - Answer-A square matrix that possesses an inverse
QUESTIONS WITH CORRECT ANSWERS
Consistent System - Answer-a linear system with at least one solution
inconsistent system - Answer-a linear system with no solution
linearly independent vectors - Answer-An indexed set {v1 , ... , vp} with the property that
the vector equation c1 v1 + c2 v2 + · · · + cp vp = 0 has only the trivial solution,
c1=···=cp=0.
linearly dependent vectors - Answer-An indexed set {v1, ... , vp} with the property that
there exist weights c1,•••,cp, not all zero, such that c1 v1 + · · · + cp vp = 0. That is, the
vector equation c1 v1 + c2 v2 + · · · + cp vp = 0 has a nontrivial solution.
Domain (of a transformation T: R"→R"') - Answer-The set of all vectors x for which T(x)
is defined (R"')
Codomain (of a transformation T: R"→R"') - Answer-The set Rm that contains the range
of T. In general, if T maps a vector space V into a vector space W, then W is called the
codomain of T (R")
Range (of a transformation T: R"→R"') - Answer-The set of all vectors (b) that live in the
codomain such that there is a vector (x) in Rm with T(x)=b
Linear Transformation - Answer-A rule T that assigns to each vector x in V a unique
vector T(x) in W, such that...
(i) T(u + v) = T(u) + T(v) for all u, v in V
(ii) T(cu) = cT(u) for all u in V and all scalars c.
Notation: T: V→W; also, x→Ax when T: R"→R"' and A is the standard matrix for T.
one-to-one transformation - Answer-A mapping T: R"→R"' such that each b in R"' is the
image of at most one x in R"
Onto Transformation - Answer-A mapping T: R"→R"' such that each b in R"' is the
image of at least one x in R"
Invertible Transformation - Answer-A linear transformation T: R"→R" such that there
exists a function S: R"→R" satisfying both T(S(x))=x and S(T(x))=x for all x in R"
Invertible (Nonsingular) Matrix - Answer-A square matrix that possesses an inverse
