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Worksheet: Basis and Dimension of a Given Span

Q1:

Determine a basis and the dimension of the given span.

  • A basis = 1 2 0 , 1 3 1 , 0 1 1 , dimension = 3
  • B basis = 1 2 0 , 2 4 0 , dimension = 2
  • C basis = 1 2 0 , 2 4 0 , 1 3 1 , dimension = 3
  • D basis = 1 2 0 , 1 3 1 , dimension = 2
  • E basis = 1 2 0 , 2 4 0 , 2 6 2 , dimension = 3

Q2:

Fill in the blank. The of 𝑚 orthogonal vectors is 𝑚 -dimensional.

  • Akernel
  • Bcollection
  • Ctransformation
  • Dspan

Q3:

Determine a basis and the dimension of the given span

  • A basis = 1 4 0 , 1 2 0 , 2 6 2 , 0 1 1 , dimension = 4
  • B basis = 1 2 0 , 0 1 1 , dimension = 2
  • C basis = 1 2 0 , 0 1 1 , 1 3 1 , dimension = 3
  • D basis = 1 2 0 , 1 4 0 , 1 3 1 , dimension = 3
  • E basis = 1 2 0 , 1 4 0 , 0 1 1 , 1 3 1 , dimension = 4

Q4:

Determine whether the three vectors are linearly independent or linearly dependent.

  • ALinearly Independent
  • BLinearly Dependent

Q5:

Determine a basis and the dimension of the span

  • ABasis = 1 2 , 2 4 , dimension = 3
  • BBasis = 1 2 , 2 4 , dimension = 2
  • CBasis = 1 2 , 1 3 , dimension = 3
  • DBasis = 1 2 , 1 3 , dimension = 2
  • EBasis = 2 6 , 2 4 , dimension = 2

Q6:

Determine whether the three given vectors are linearly independent or linearly dependent.

  • Alinearly independent
  • Blinearly dependent

Q7:

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

  • ATrue
  • BFalse

Q8:

Determine whether the three vectors are linearly independent or linearly dependent.

  • ALinearly dependent
  • BLinearly independent