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 12,24,13.

  • ABasis=26,24, dimension =2
  • BBasis=12,24, dimension =2
  • CBasis=12,13, dimension =2
  • DBasis=12,13, dimension =3
  • EBasis=12,24, dimension =3

Q2:

Determine a basis and the dimension of the given span 120,140,131,011.

  • Abasis =120,140,131, dimension =3
  • Bbasis =120,011,131, dimension =3
  • Cbasis =120,011, dimension =2
  • Dbasis =120,140,011,131, dimension =4
  • Ebasis =140,120,262,011, dimension =4

Q3:

Determine a basis and the dimension of the given span. 120,240,131,011

  • Abasis=120,240, dimension=2
  • Bbasis=120,240,262, dimension=3
  • Cbasis=120,240,131, dimension=3
  • Dbasis=120,131,011, dimension=3
  • Ebasis=120,131, dimension=2

Q4:

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

  • AFalse
  • BTrue

Q5:

Determine whether the given vectors are a basis for , and state whether or not they span it 103,433,120,240.

  • 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 103,010,120 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 103,010,120,000 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 103,010,113,000 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 2𝑡+3𝑠𝑠𝑡𝑡+𝑠𝑠,𝑡.

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

  • ANo, no basis.
  • BYes, basis: 211,511, dimension: 2.
  • CYes, basis: 211,311, dimension: 2.
  • DNo, basis: 211,311, dimension: 1.
  • EYes, basis: 211,511, dimension: 2.

Q10:

Consider the vectors in the form 2𝑡+3𝑠+𝑢𝑠𝑡𝑡+𝑠𝑢𝑠,𝑡,𝑢.

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

  • Ayes, basis: 3110,2110,1001, dimension: 3
  • Byes, basis: 3111,2110,1001, dimension: 3
  • Cyes, basis: 3111,2110,1001, dimension: 3
  • Dyes, basis: 3110,2110,1001, dimension: 3
  • Eno, no basis and no dimension

Q11:

Consider the set of vectors 2𝑡+𝑢𝑡+3𝑢𝑡+𝑠+𝑣𝑢𝑠,𝑡,𝑢,𝑣

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

  • AYes, basis: 0010,2111,1301, dimension: 3.
  • BYes, basis: 0010,2110,1311, dimension: 3.
  • CNo
  • DYes, basis: 0010,2110,1301, dimension: 3.
  • EYes, basis: 0011,2110,1301, dimension: 3.

Q12:

Let 𝑉 be the space of polynomials in the variable 𝑥 that have degree less than 4. Is 𝑥+1,𝑥+𝑥,2𝑥+𝑥,2𝑥𝑥3𝑥+1 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 𝑉 be the space of polynomials in the variable 𝑥 that have degree less than 4. Is 𝑥+1,𝑥+𝑥+2𝑥,𝑥+𝑥,𝑥+𝑥+𝑥 a basis for this space?

  • Ano
  • Byes

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