©THEBRIGHT EXAM SOLUTIONS
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Math 110 Exam Questions And Answers
100% Pass
Linear combination (2.3) - answer✔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 - answer✔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.... - answer✔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 - 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
, ©THEBRIGHT EXAM SOLUTIONS
11/8/2024 12:08 PM
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) - answer✔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 - answer✔A vector space is called infinite-dimensional if it is not finite-
dimensional
linearly independent - answer✔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.
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: - 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
equal too
less than and equal too - answer✔Length of linearly independent list less than and equal toothe length
of spanning list
