# Worksheet: Basis of a Vector Space

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