MATH 110 MODULE 1 |EXAM 2025/2026 |NEW UPDATED QUESTIONS & ANSWERS |GRADED A|100% CORRECT

MATH 110 MODULE 1 |EXAM 2025/2026 |NEW UPDATED QUESTIONS &
ANSWERS |GRADED A|100% CORRECT


1.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.



2.Matrix of a nilpotent operator - ANSWER all entries on and below the diagonal are 0's.



3. multiplicity - ANSWER Suppose T in L(V). The multiplicity of an eigenvalue λ of T is defined to be the
dimension of the corresponding generalized eigenspace G(λ,T)



4. the multiplicity of an eigenvalue λ of T=? - ANSWER = dimnull(T-λI)^dimV



5. Then the sum of the multiplicities of all the eigenvalues of T=? - ANSWER Then the sum of the
multiplicities of all the eigenvalues of T equals dim V.



6. block diagonal matrix - ANSWER a partitioned matrix with zero blocks off the main diagonal (of
blocks). such a matrix is invertible if and only if each block on the diagonal is invertible.



7. characteristic polynomial - ANSWER Suppose V is a complex vector space.

and T in L(V)

Let λ1...λm denote the distinct eigenvalues of T, with multiplicities d1,..,dm. The charateristic polynomial
is (z-λ1)^d1 ...(z-λm)^dm



8. 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}



9. 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.



10. spans - ANSWER If span(v1,....,vm) equals V, we say that v1,...,vm spans V.



11. finite-dimensional vector space - ANSWER A vector space is called finite-dimensional if some list of
vectors in it spans the space.



12. 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.



13. 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

p(z)= a0+a1z+...+amz^m

for all z in F.

-If p has degree m, we write

degp =m.



14. 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.



15. infinite-dimensional vector space - ANSWER A vector space is called infinite-dimensional if it is not
finite-dimensional

, 16. 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.



17. 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.



18. 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)



19. 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



20.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 - ANSWER Every subspace of a finite-dimensional
vector space is finite-dimensional



21. basis - ANSWER A basis of V is a list of vectors in V that is linearly independent and spans V.



22.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.