Worksheet: Gram–Schmidt Process

In this worksheet, we will practice finding the orthonormal basis of a vector space using the Gram–Schmidt process.

Q1:

Find an orthonormal basis of eigenvectors for the matrix 3000321201232.

  • A 1 0 0 , 1 2 0 1 1 , 1 2 0 1 1
  • B 1 0 0 , 0 2 1 , 0 1 2
  • C 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1
  • D 1 0 0 , 1 5 0 2 1 , 1 5 0 1 2
  • E 1 0 0 , 0 1 1 , 0 1 1

Q2:

Find an orthonormal basis of eigenvectors for the matrix 200051015.

  • A 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1
  • B 1 0 0 , 0 1 1 , 0 1 1
  • C 1 0 0 , 1 5 0 2 1 , 1 5 0 1 2
  • D 1 0 0 , 0 2 1 , 0 1 2
  • E 1 0 0 , 1 2 0 1 1 , 1 2 0 1 1

Q3:

Find an orthonormal basis of eigenvectors for the matrix 177471744414.

  • A 1 1 1 , 1 0 1 , 1 2 1
  • B 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1
  • C 1 3 1 1 1 , 1 6 1 1 2 , 1 2 1 1 0
  • D 1 3 1 1 1 , 1 6 1 1 2 , 1 2 1 1 0
  • E 1 1 1 , 1 1 2 , 1 1 0

Q4:

Find an orthonormal basis of eigenvectors for the matrix 111411144414.

  • A 1 3 1 1 1 , 1 2 1 1 0 , 1 6 1 1 2
  • B 1 1 1 , 1 0 1 , 1 2 1
  • C 1 1 1 , 1 1 0 , 1 1 2
  • D 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1
  • E 1 3 1 1 1 , 1 2 1 1 0 , 1 6 1 1 2

Q5:

Given that 𝐴=131411344410, find the eigenvalues and an orthonormal basis of eigenvectors for 𝐴.

  • Aeigenvalues: 2, 6, and 12, eigenvectors: 161122,121106,1311112
  • Beigenvalues: 6, 12, and 18, eigenvectors: 161126,1211012,1311118
  • Ceigenvalues: 2, 6, and 12, eigenvectors: 161122,1211012,131116
  • Deigenvalues: 6, 12, and 18, eigenvectors: 1611212,121106,1311118
  • Eeigenvalues: 6, 12, and 18, eigenvectors: 161126,1211012,1311118

Q6:

Find an orthonormal basis of eigenvectors for the matrix 533015851530151456158515615715.

  • A 1 0 5 , 1 0 3 2 , 5 2 4 1
  • B 1 3 1 1 1 , 1 2 1 0 1 , 1 6 1 2 1
  • C 1 6 1 0 5 , 3 0 1 5 5 6 2 1 , 1 3 0 5 2 6 1
  • D 1 2 6 1 0 5 , 1 1 1 3 1 0 3 2 , 1 6 0 2 5 2 4 1
  • E 1 0 5 , 5 6 2 1 , 5 2 6 1

Q7:

The two level surfaces 2𝑥+3𝑦𝑧+𝑤=0 and 3𝑥𝑦+𝑧+2𝑤=0 intersect in a subspace of . Find a basis for this subspace, and then find an orthonormal basis for this subspace.

  • ABasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209
  • BBasis: 25110, 71011, orthonormal basis: 1661156113060, 463,135620911,25462091330620952096209
  • CBasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209
  • DBasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209
  • EBasis: 25110, 71011, orthonormal basis: 1156166113060, 463,135620911,25462091330620952096209

Q8:

Apply the Gram–Schmidt process to the vectors (1,2,1), (2,1,3), and (1,0,0) to find an orthonormal basis for their span.

  • A 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • B 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • C 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • D 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3
  • E 6 6 6 3 6 6 , 3 2 1 0 2 2 5 2 2 , 7 3 1 5 3 1 5 3 3

Q9:

Apply the Gram–Schmidt process to the vectors (3,4,0), (7,1,0), and (1,7,1) to find an orthonormal basis for their span.

  • A 4 5 3 5 0 , 3 5 4 5 0 , 0 0 1
  • B 4 5 3 5 0 , 4 5 3 5 0 , 0 0 1
  • C 3 5 4 5 0 , 3 5 4 5 0 , 0 0 1
  • D 3 5 4 5 0 , 3 5 4 5 0 , 0 0 1
  • E 3 5 4 5 0 , 4 5 3 5 0 , 0 0 1

Q10:

Apply the Gram–Schmidt process to the vectors (1,2,1,0), (2,1,3,1), and (1,0,0,1) to find an orthonormal basis for their span.

  • A 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • B 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • C 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • D 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3
  • E 6 6 6 3 6 6 0 , 6 6 2 6 9 5 6 1 8 6 9 , 5 1 1 1 1 1 1 1 1 1 3 3 3 1 7 1 1 1 3 3 3 2 2 1 1 1 3 3 3

Q11:

Which of the following is an orthonormal set of vectors in ?

  • A 2 0 0 , 0 1 0 , 0 0 3
  • B 1 3 1 3 1 3 , 1 2 0 1 2
  • C 1 1 1 , 1 0 1
  • D 0 1 0 , 0 0 1 , 0 0 0

Q12:

The set 𝑉={(𝑥,𝑦,𝑧)2𝑥+3𝑦𝑧=0} is a subspace of . Find an orthonormal basis for this subspace.

  • A 2 5 5 0 5 5 , 3 7 0 3 5 7 0 1 4 3 7 0 7 0
  • B 2 5 5 0 5 5 , 3 7 0 3 5 7 0 1 4 3 7 0 7 0
  • C 2 5 5 0 6 6 , 7 0 3 5 7 0 1 4 7 0 7 0
  • D 5 5 0 2 5 5 , 3 7 0 3 5 7 0 1 4 3 7 0 7 0
  • E 5 5 0 2 5 5 , 3 7 0 3 5 7 0 1 4 3 7 0 7 0

Q13:

Fill in the blank. Columns of an 𝑛×𝑛 matrix 𝐴 are an orthonormal basis for , if and only if 𝐴 is a matrix.

  • Anormal
  • Bunitary
  • Csymmetric
  • Dsquare

Q14:

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

  • Acollection
  • Bspan
  • Ckernel
  • Dtransformation

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