Question 1

Find an orthogonal basis for the subspace  .

Accepted Answer

The answer of Find an orthogonal basis for the subspace...

Question 2

Find  , where  .

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The answer of Find  , where  ....

Question 3

Apply the Gram-Schmidt process to the given basis for  to produce an orthogonal basis for  . Then normalize the vectors to produce an orthonormal basis for  .

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The answer of Apply the Gram-Schmidt process to the given...

Question 4

Find an orthonormal basis for the subspace  .

Accepted Answer

The answer of Find an orthonormal basis for the subspace...

Question 5

Find an orthonormal basis for the subspace  , and use it to to find  , where  .

Accepted Answer

The answer of Find an orthonormal basis for the subspace...

Question 6

If S is a nonzero subspace of  , then for every vector u in  ,  belongs to S.

Accepted Answer

The answer of If S is a nonzero subspace of...

Question 7

If  is a nonzero subspace of  , and u belongs to  , then  .

Accepted Answer

Since the given subspace is nonzero, it contains the zero vector. Therefore, by definition of subspace, it is closed under addition and scalar multiplication. Since u belongs to a larger subspace, it also belongs to the smaller subspace. Therefore, the given statement is true.

Question 8

If  is a nonzero subspace of  , and u belongs to  , then  for every vector  .

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The answer of If  is a nonzero subspace of...

Question 9

If  and  are nonzero subspaces of  , then  for every vector u in  .

Accepted Answer

Since both 11edaba5_20a1_edc6_a975_bbdae2f9cc0e_TBMC1039_00 and 11edaba5_20a1_edc7_a975_f72d2f2451aa_TBMC1039_00 are nonzero subspaces of 11edaba5_20a1_edc8_a975_33d05a431915_TBMC1039_00, they both contain the zero vector. Therefore, for any vector u in 11edaba5_20a1_edca_a975_35699b9a02db_TBMC1039_00, the linear combination of u and the zero vector, which is zero, will also be in both subspaces. Hence, their intersection is nonzero and contains the zero vector, which means that the entire space 11edaba5_20a1_edc9_a975_7b0ce8886788_TBMC1039_00 is contained in their sum. Therefore, for any vector u in 11edaba5_20a1_edca_a975_35699b9a02db_TBMC1039_00, it will also be in 11edaba5_20a1_edc9_a975_7b0ce8886788_TBMC1039_00.

Question 10

If  is an orthonormal set, then  is linearly independent.

Accepted Answer

If the first set is orthonormal, then all its vectors must be unit vectors and orthogonal to each other. Adding a scalar multiple of one vector to another cannot change the orthogonality of the set, so the second set must also be orthogonal. Furthermore, since the second vector in the second set is not a scalar multiple of the first vector, the set is linearly independent. Therefore, the statement is true.

Question 11

Determine if the given matrix is symmetric.

Accepted Answer

The answer of Determine if the given matrix is symmetric.
...

Question 12

Determine if the given matrix is orthogonal.

Accepted Answer

The answer of Determine if the given matrix is orthogonal.
...

Question 13

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Accepted Answer

The answer of The eigenvalues and corresponding eigenvectors for a...

Question 14

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Accepted Answer

The answer of The eigenvalues and corresponding eigenvectors for a...

Question 15

The eigenvalues for the symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Accepted Answer

The answer of The eigenvalues for the symmetric matrix A...

Question 16

The eigenvalues for the symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Accepted Answer

The answer of The eigenvalues for the symmetric matrix A...

Question 17

Verify that the eigenvalues of  are nonnegative.

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The answer of Verify that the eigenvalues of  are...

Question 18

Determine  from the given matrix with orthogonal columns without using row operations.

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The answer of Determine  from the given matrix with...

Question 19

Find the QR decomposition for the matrix  .

Accepted Answer

The answer of Find the QR decomposition for the matrix...

Question 20

Find the QR decomposition for the matrix  .

Accepted Answer

The answer of Find the QR decomposition for the matrix...

Quiz 8: Orthogonality

Find an orthogonal basis for the subspace .

Find , where .

Apply the Gram-Schmidt process to the given basis for to produce an orthogonal basis for . Then normalize the vectors to produce an orthonormal basis for .

Find an orthonormal basis for the subspace .

Find an orthonormal basis for the subspace , and use it to to find , where .

If S is a nonzero subspace of , then for every vector u in , belongs to S.

If is a nonzero subspace of , and u belongs to , then .

If is a nonzero subspace of , and u belongs to , then for every vector .

If and are nonzero subspaces of , then for every vector u in .

If is an orthonormal set, then is linearly independent.

Determine if the given matrix is symmetric.

Determine if the given matrix is orthogonal.

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

The eigenvalues for the symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

The eigenvalues for the symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Verify that the eigenvalues of are nonnegative.

Determine from the given matrix with orthogonal columns without using row operations.

Find the QR decomposition for the matrix .

Find the QR decomposition for the matrix .