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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⎤⎥⎥⎥⎥⎦.

Q2:

Find an orthonormal basis of eigenvectors for the matrix 200051015.

Q3:

Find an orthonormal basis of eigenvectors for the matrix 17−7−4−717−4−4−414.

Q4:

Find an orthonormal basis of eigenvectors for the matrix 11−1−4−111−4−4−414.

Q5:

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

Q6:

Find an orthonormal basis of eigenvectors for the matrix ⎡⎢⎢⎢⎢⎢⎢⎢⎣−53√30158√515√3015−145−√6158√515−√615715⎤⎥⎥⎥⎥⎥⎥⎥⎦.

Q7:

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

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.

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.

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.

Q11:

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

Q12:

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

Q13:

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

Q14:

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

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