In this worksheet, we will practice identifying whether a set of vectors forms a basis for a certain vector space and defining the dimension of a vector space (or subspace).

Q1:

Determine a basis and the dimension of the span

ABasis, dimension

BBasis, dimension

CBasis, dimension

DBasis, dimension

EBasis, dimension

Q2:

Determine a basis and the dimension of the given span

Abasis ,
dimension

Bbasis ,
dimension

Cbasis ,
dimension

Dbasis ,
dimension

Ebasis ,
dimension

Q3:

Determine a basis and the dimension of the given span.

Abasis,
dimension

Bbasis,
dimension

Cbasis,
dimension

Dbasis,
dimension

Ebasis,
dimension

Q4:

True or False: In a vector space of dimension , any spanning set of
cardinality is a basis.

AFalse

BTrue

Q5:

Determine whether the given vectors are a basis for ,
and state whether or not they span it

AThey are a basis for and they span
.

BThey are not a basis for and they span
.

CThey are not a basis for and they do not span
.

DThey are a basis for and they do not span
.

Q6:

Determine whether the vectors
are a basis for and state whether they span it.

AThey are a basis for ,
and they do not span .

BThey are a basis for ,
and they span .

CThey are not a basis for ,
and they span .

DThey are not a basis for ,
and they do not span .

Q7:

Determine whether the vectors
are a basis for , and state whether they span it.

AThey are not a basis for , and they span .

BThey are not a basis for , and they do not span .

CThey are a basis for , and they span .

DThey are a basis for , and they do not span .

Q8:

Determine whether the vectors
are a basis for , and state whether they span it.

AThey are not a basis for , and they span
.

BThey are a basis for , and they do not span
.

CThey are not a basis for , and they do not span
.

DThey are a basis for , and they span
.

Q9:

Consider the vectors

Is this set of vectors a subspace of ? Determine a basis for the subspace and its dimension.

ANo, no basis.

BYes, basis:
, dimension:
2.

CYes, basis:
, dimension:
2.

DNo, basis:
, dimension:
1.

EYes, basis:
, dimension:
2.

Q10:

Consider the vectors in the form

Is this set of vectors a subspace of ?
Find a basis for this subspace and find its dimension.

Ayes, basis:
, dimension: 3

Byes, basis:
, dimension: 3

Cyes, basis: , dimension: 3

Dyes, basis: , dimension: 3

Eno, no basis and no dimension

Q11:

Consider the set of vectors

Is this set of vectors a subspace of ? If yes, determine a basis for the subspace and its dimension.

AYes, basis:
, dimension: 3.

BYes, basis:
, dimension: 3.

CNo

DYes, basis:
, dimension: 3.

EYes, basis:
, dimension: 3.

Q12:

Let be the space of polynomials in the variable that have degree less than 4.
Is a basis for this space?

Ano

Byes

Q13:

True or False: Any set of 4 vectors in a 3-dimensional vector space must be linearly dependent.

AFalse

BTrue

Q14:

Let be the space of polynomials in the variable that have degree less than 4.
Is
a basis for this space?

Ano

Byes

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