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Math 110 Exam Questions And Accurate Answers

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Math 110 Exam Questions And Accurate Answers...

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  • November 2, 2024
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  • Exam (elaborations)
  • Questions & answers
  • Math 110
  • Math 110
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Chrisyuis
Math 110 Exam Questions And Accurate Answers


linear 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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