Math 110 Exam questions and answers 2025
linear combination (2.3)
A linear combination of a list v1,....,vm of vectors in V is a vector of the form:
a1v1+....+amvm
where a1,...,am are in F.
Span
The set of all linear combinations of a list of vectors v1,....,vm in V is
called the
span of v1,....,vm,
denoted span(v1,.....,vm).
In other words,
span(v1,....,vm) = {a1v1+...+amvm : a1,....,am}
The span of a list of vectors in V is the smallest....
Span is the smallest containing subspace.
The span of a list of vectors in V is the smallest subspace of V containing all the vectors in the
list.
spans
If span(v1,....,vm) equals V, we say that v1,...,vm spans V.
finite-dimensional vector space
A vector space is called finite-dimensional if some list of vectors in it spans the space.
Polynomial P(F)
A function p:F->F is called a polynomial with coefficients in F
if there exist a0,...,am in F such that
p(z)= a0+a1z+a2z^2+.....+ amz^m
,for all z in F.
P(F) is the set of all polynomials with coefficients in F.
degree of a polynomial, deg p
A polynomial p in P(F) is said to have degree m if there exist
scalars a0,a1,...,am in F with am not equal 0 such that
p(z)= a0+a1z+...+amz^m
for all z in F.
-If p has degree m, we write
degp =m.
Pm(F)
For m a nonnegative integer, Pm(F) denotes the set of all polynomials with coefficients in F and
degree at most m.
infinite-dimensional vector space
A vector space is called infinite-dimensional if it is not finite-dimensional
linearly independent
A list v1,....,vm of vectors in V is called linearly independent if the only choice of a1,...,am in F
that makes
a1v1+...+amvm= 0 is a1=....=am=0
The empty list {0} is also declared to be linearly independent.
linearly dependent
A list of vectors in V is called linearly dependent if it is not linearly independent.
In other words, a list v1,...,vm of vectors in V is linearly dependent if there exist a1,...,am in F,
not all 0, such that a1v1+...+amvm=0.
Suppose v1,...,vm is a linearly dependent list in V. Then there exists j in {1,2,...m} such that
the following hold:
, (a) vj is in span(v1,...,vj-1)
(b) if the jth term is removed from v1,...,vm, the span of the remaining list equals
span(v1,...,vm)
Length of linearly independent list is______________ the length of spanning list
less than
greater than
equal too
less than and equal too
Length of linearly independent list less than and equal toothe length of spanning list
In a finite-dimensional vector space, the length of every linearly independent list of vectors is
less than or equal to the length of every spanning list of vectors.
Every subspace of a finite-dimensional vector space is
Every subspace of a finite-dimensional vector space is finite-dimensional
basis
A basis of V is a list of vectors in V that is linearly independent and spans V.
Criterion for basis
A list v1,...,vn of vectors in V is a basis of V if and only if every v in V can be written uniquely in
the form:
v = a1v1+...+anvn
where a1,...,an is in F.
Every spanning list in a vector space can be reduced...
linear combination (2.3)
A linear combination of a list v1,....,vm of vectors in V is a vector of the form:
a1v1+....+amvm
where a1,...,am are in F.
Span
The set of all linear combinations of a list of vectors v1,....,vm in V is
called the
span of v1,....,vm,
denoted span(v1,.....,vm).
In other words,
span(v1,....,vm) = {a1v1+...+amvm : a1,....,am}
The span of a list of vectors in V is the smallest....
Span is the smallest containing subspace.
The span of a list of vectors in V is the smallest subspace of V containing all the vectors in the
list.
spans
If span(v1,....,vm) equals V, we say that v1,...,vm spans V.
finite-dimensional vector space
A vector space is called finite-dimensional if some list of vectors in it spans the space.
Polynomial P(F)
A function p:F->F is called a polynomial with coefficients in F
if there exist a0,...,am in F such that
p(z)= a0+a1z+a2z^2+.....+ amz^m
,for all z in F.
P(F) is the set of all polynomials with coefficients in F.
degree of a polynomial, deg p
A polynomial p in P(F) is said to have degree m if there exist
scalars a0,a1,...,am in F with am not equal 0 such that
p(z)= a0+a1z+...+amz^m
for all z in F.
-If p has degree m, we write
degp =m.
Pm(F)
For m a nonnegative integer, Pm(F) denotes the set of all polynomials with coefficients in F and
degree at most m.
infinite-dimensional vector space
A vector space is called infinite-dimensional if it is not finite-dimensional
linearly independent
A list v1,....,vm of vectors in V is called linearly independent if the only choice of a1,...,am in F
that makes
a1v1+...+amvm= 0 is a1=....=am=0
The empty list {0} is also declared to be linearly independent.
linearly dependent
A list of vectors in V is called linearly dependent if it is not linearly independent.
In other words, a list v1,...,vm of vectors in V is linearly dependent if there exist a1,...,am in F,
not all 0, such that a1v1+...+amvm=0.
Suppose v1,...,vm is a linearly dependent list in V. Then there exists j in {1,2,...m} such that
the following hold:
, (a) vj is in span(v1,...,vj-1)
(b) if the jth term is removed from v1,...,vm, the span of the remaining list equals
span(v1,...,vm)
Length of linearly independent list is______________ the length of spanning list
less than
greater than
equal too
less than and equal too
Length of linearly independent list less than and equal toothe length of spanning list
In a finite-dimensional vector space, the length of every linearly independent list of vectors is
less than or equal to the length of every spanning list of vectors.
Every subspace of a finite-dimensional vector space is
Every subspace of a finite-dimensional vector space is finite-dimensional
basis
A basis of V is a list of vectors in V that is linearly independent and spans V.
Criterion for basis
A list v1,...,vn of vectors in V is a basis of V if and only if every v in V can be written uniquely in
the form:
v = a1v1+...+anvn
where a1,...,an is in F.
Every spanning list in a vector space can be reduced...
