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Math 110 Exam Questions And Accurate Answers
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Math 110 Exam Questions And Accurate Answerslinear combination (2.3) - Solution 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 - Solution 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
Span is the smallest containing subspace.
Span of list of vectors in V is the smallest subspace of V containing all the vectors in the
list.
spans - Answer If span(v1,.,vm) equals V, we say that v1,.,vm spans V.
finite-dimensional vector space - Answer A vector space is called finite-dimensional if
some list of vectors in it spans the space.
Polynomial P(F) - Answer 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 - Answer 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
np(z)= a0+a1z+.+amz^m
for all z in F.
-If p has degree m, we write
degp =m.
Pm(F) - Solution 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 - Solution A vector space is called infinite-dimensional
if it is not finite-dimensional
linearly independent - Solution 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 - Answer A list of vectors in V is called linearly dependent if it is not
linearly independent.
That is, 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: - Answer (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
=too
less than and equal too - Answer Length of linearly independent list less than and equal
toothe length of spanning list
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 - Answer Every subspace of a
finite-dimensional vector space is finite-dimensional
basis - Answer A basis of V is a list of vectors in V that is linearly independent and spans
V.
Criterion for basis - Answer 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. - Answer Spanning list contains a
basis
Every spanning list in a vector space can be reduced to a basis of the vector space.
Every finite-dimensional vector space has. - Answer Basis of finite-dimensional vector
space
Every finite-dimensional vector space has a basis.
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