Master the Gram-Schmidt Process: Orthogonalize Vectors with Ease

Question 1

In the Gram-Schmidt process, how do you obtain vector v_k after k steps?

Accepted Answer

Subtract the projection of v_k onto the subspace spanned by the previous vectors, then normalize

Answer

Add the projection of v_k onto the subspace spanned by the previous vectors, then normalize

Answer

Normalize v_k

Answer

Subtract v_1 from v_k

Question 2

Using the Gram-Schmidt process, find an orthonormal basis for the subspace spanned by the vectors (1, 1, 1), (2, 1, 0), and (3, 1, 1). Express your answer in the form {u₁, u₂, u₃} = {q₁, q₂, q₃}.

Accepted Answer

{(1, 1/√3, 1/√3), (0, 2/√6, -1/√6), (0, 0, 1)}

Answer

{(1, 1, 1), (2, 1, 0), (3, 1, 1)}

Answer

{(1, 0, 0), (0, 1, 0), (0, 0, 1)}

Question 3

What is the orthogonal projection of the vector (2, 3) onto the vector (1, -1)?

Accepted Answer

(2/5, -3/5)

Answer

(-1, 1)

Answer

(1, 1)

Answer

(2, -2)

Question 4

Which of the following is NOT a subspace?

Accepted Answer

The set of all vectors with zero length

Answer

The set of all vectors perpendicular to the vector (1, 0)

Answer

The set of all vectors parallel to the xy-plane

Answer

The set of all vectors lying on a plane

Question 5

The Gram-Schmidt process can be used to find the inverse of which matrix type?

Accepted Answer

Orthogonal matrix

Answer

Symmetric matrix

Answer

Diagonal matrix

Answer

Triangular matrix

Question 6

What is a key difference between the Gram-Schmidt process and the QR decomposition?

Accepted Answer

The Gram-Schmidt process results in a set of orthogonal vectors, while the QR decomposition results in a set of orthonormal vectors

Answer

The QR decomposition can be used to find the eigenvalues of a matrix, while the Gram-Schmidt process cannot

Answer

The Gram-Schmidt process can be applied to any set of vectors, while the QR decomposition is only applicable to linearly independent sets

Answer

The Gram-Schmidt process is an algebraic method, while the QR decomposition is a numerical method

Question 7

What is the first step in the Gram-Schmidt process, which aims to obtain an orthonormal basis from a set of vectors?

Accepted Answer

Normalize the first vector in the set.

Answer

Compute the orthogonal complement of the first vector with respect to the subspace spanned by the remaining vectors.

Answer

Divide each vector by its length.

Answer

Project each vector onto the subspace spanned by the previous vectors.

Question 8

What is the expected outcome of applying the Gram-Schmidt process to a set of linearly independent vectors?

Accepted Answer

An orthonormal basis for the subspace spanned by the original vectors.

Answer

A set of orthogonal vectors that are not necessarily of unit length.

Answer

A set of linearly independent vectors that are not orthogonal.

Answer

A set of linearly dependent vectors that span a smaller subspace.

Question 9

Which of the following is NOT a potential benefit of using the Gram-Schmidt process?

Accepted Answer

Orthogonalizing a set of vectors that are already orthonormal.

Answer

Solving systems of linear equations.

Answer

Finding eigenvalues and eigenvectors.

Answer

Reducing the computational complexity of certain algorithms.

Question 10

Consider a set of three linearly independent vectors in R^3. How many orthogonal vectors will the Gram-Schmidt process produce?

Accepted Answer

3

Answer

1

Answer

2

Answer

None

Question 11

How is the Gram-Schmidt process related to orthogonal projections?

Accepted Answer

It is equivalent to finding the orthogonal projection of each vector onto the subspace spanned by the previous vectors.

Answer

It constructs orthogonal vectors using orthogonal projections.

Answer

It is not related to orthogonal projections.

Answer

It uses orthogonal projections to find the null space of a matrix.

Question 12

Which of the following is the first step in the Gram-Schmidt process for a set of vectors in an inner product space?

Accepted Answer

Normalize the first vector

Answer

Subtract the projection of each vector onto all previous vectors

Answer

Find the orthogonal complement of the subspace spanned by the vectors

Answer

Gram-Schmidt orthogonalization is not a multi-step process

Question 13

What property do the vectors obtained from the Gram-Schmidt process have?

Accepted Answer

They are linearly independent and orthonormal

Answer

They are orthogonal but not linearly independent

Answer

They are linearly independent but not orthonormal

Answer

Neither linearly independent nor orthonormal

Question 14

Which of the following statements about orthogonal matrices is correct?

Accepted Answer

An orthogonal matrix is a square matrix whose inverse is equal to its transpose

Answer

An orthogonal matrix is a rectangular matrix whose determinant is 1

Answer

An orthogonal matrix is a square matrix whose eigenvalues are all real

Answer

An orthogonal matrix is a diagonal matrix

Question 15

What is the relationship between the eigenvalues and eigenvectors of a symmetric matrix?

Accepted Answer

Eigenvalues are real and eigenvectors are orthogonal

Answer

Eigenvalues are complex and eigenvectors are linearly independent

Answer

Eigenvalues are real and eigenvectors are linearly dependent

Answer

Eigenvalues are complex and eigenvectors are orthogonal

Question 16

Which of the following transformations preserves the orthogonality of a set of vectors?

Accepted Answer

Rotation

Answer

Reflection

Answer

Scaling

Answer

Shearing

Question 17

Which of the following equations represents the Gram-Schmidt process used for orthogonalizing a set of vectors in an inner product space?

Accepted Answer

x_i = x_i - sum( (x_i dot x_j) / (x_j dot x_j) ) * x_j, for j < i

Answer

x_i = x_i + sum( (x_i dot x_j) / (x_j dot x_j) ) * x_j, for j < i

Answer

x_i = x_i - sum( (x_j dot x_i) / (x_j dot x_j) ) * x_j, for j < i

Answer

x_i = x_i + sum( (x_j dot x_i) / (x_j dot x_j) ) * x_j, for j < i

Question 18

What is the primary purpose of the Gram-Schmidt process?

Accepted Answer

To orthogonalize a set of vectors in an inner product space

Answer

To find the eigenvalues of a matrix

Answer

To solve systems of linear equations

Answer

To compute the determinant of a matrix

Question 19

What is the mathematical operation that computes the inner product of two vectors?

Accepted Answer

Dot product

Answer

Cross product

Answer

Sum

Answer

Difference

Question 20

Which of the following sets of vectors forms an orthonormal basis for R^2?

Accepted Answer

{ (1,0), (0,1) }

Answer

{ (1,1), (0,1) }

Answer

{ (1,0), (1,1) }

Answer

{ (0,1), (1,0) }

Question 21

How is the Gram-Schmidt process related to orthogonal matrices?

Accepted Answer

The Gram-Schmidt process can be used to construct orthogonal matrices

Answer

Orthogonal matrices can be used to perform the Gram-Schmidt process

Answer

The Gram-Schmidt process and orthogonal matrices are unrelated

Answer

The Gram-Schmidt process can be used to decompose orthogonal matrices

Question 22

In the context of linear regression, what is the role of the Gram-Schmidt process?

Accepted Answer

Orthogonalizing the predictor variables

Answer

Finding the least squares solution

Answer

Computing the regression coefficients

Answer

Calculating the variance of the residuals

Question 23

Which of the following is a potential disadvantage of using the Gram-Schmidt process?

Accepted Answer

Can be computationally expensive.

Answer

Produces an orthonormal basis.

Answer

Requires the vectors to be linearly dependent.

Question 24

State the primary purpose of the Gram-Schmidt process:

Accepted Answer

To construct an orthonormal basis for a subspace

Answer

To find the eigenvalues of a matrix

Answer

To solve systems of linear equations

Answer

To determine the rank of a matrix

Question 25

Which property best characterizes an orthonormal basis?

Accepted Answer

The vectors are orthogonal to each other and have unit length

Answer

The vectors are orthogonal to each other but do not have unit length

Answer

The vectors are linearly independent and span the subspace

Question 26

In solving systems of linear equations, the Gram-Schmidt process is utilized to:

Accepted Answer

Transform the system into an equivalent system with an upper triangular coefficient matrix

Answer

Find the general solution of the system

Answer

Determine if the system has a unique solution

Question 27

Identify a limitation of the Gram-Schmidt process:

Accepted Answer

It can be numerically unstable for ill-conditioned matrices

Answer

It can only be applied to vectors in Euclidean space

Answer

It always produces an orthonormal basis

Question 28

Describe the relationship between the Gram-Schmidt process and singular value decomposition:

Accepted Answer

The Gram-Schmidt process can be used to compute the singular value decomposition of a matrix

Answer

The Gram-Schmidt process and singular value decomposition are independent techniques

Answer

The singular value decomposition can be used to compute the Gram-Schmidt process

Question 29

Within signal processing, the Gram-Schmidt process is significant in its ability to:

Accepted Answer

Allow for the separation of signals into orthogonal components

Answer

Amplify signals

Answer

Reduce noise in signals

Question 30

Which step in the Gram-Schmidt process is performed first?

Accepted Answer

Normalize the first vector.

Answer

Find the cross product of the first two vectors.

Answer

Project the third vector onto the subspace spanned by the first two vectors.

Answer

Orthogonalize the second vector to the first vector.

Question 31

What is the primary purpose of the Gram-Schmidt process?

Accepted Answer

To construct an orthonormal basis for a subspace.

Answer

To find the eigenvalues and eigenvectors of a matrix.

Answer

To solve systems of linear equations.

Answer

To calculate the determinant of a matrix.

Question 32

Which property describes an orthonormal basis?

Accepted Answer

The vectors are mutually orthogonal and have unit length.

Answer

The vectors are linearly dependent.

Answer

The vectors span the entire vector space.

Question 33

What formula is used to orthogonalize vector v to vector u?

Accepted Answer

v - proj_u(v)

Answer

v x proj_u(v)

Answer

v + proj_u(v)

Question 34

What is a known limitation of the Gram-Schmidt process?

Accepted Answer

It is not numerically stable for ill-conditioned matrices.

Answer

It is only applicable to real vector spaces.

Answer

It cannot be used to find the null space of a matrix.

Question 35

What is the computational complexity of the Gram-Schmidt process for m vectors in an n-dimensional vector space?

Accepted Answer

O(m^2n^2)

Answer

O(mn)

Answer

O(m^3)

Answer

O(n^3)

Question 36

Which application commonly employs the Gram-Schmidt process?

Accepted Answer

QR factorization

Answer

LU decomposition

Answer

Power iteration

Answer

Gaussian elimination

Question 37

How are the Gram-Schmidt process and singular value decomposition (SVD) related?

Accepted Answer

SVD is a generalization of the Gram-Schmidt process.

Answer

SVD and Gram-Schmidt process are unrelated.

Answer

Gram-Schmidt process is a special case of the SVD.

Question 38

Which application of the Gram-Schmidt process is used to find the least squares solution to a system of linear equations?

Accepted Answer

QR factorization

Answer

Eigenvalue decomposition

Answer

Singular value decomposition

Question 39

Which of the following statements about the Gram-Schmidt process is false?

Accepted Answer

It guarantees that the resulting orthonormal basis is unique.

Answer

It always produces orthonormal vectors.

Answer

It can be used to orthogonalize any set of linearly independent vectors.

Question 40

What is the primary purpose of the Gram-Schmidt process?

Accepted Answer

To construct an orthonormal basis for a subspace from a given set of vectors.

Answer

To find the eigenvalues of a symmetric matrix.

Answer

To solve a system of linear equations directly.

Question 41

Which of the following properties is preserved by the Gram-Schmidt process for the resulting vectors?

Accepted Answer

Linear independence and span of the subspace.

Answer

Dot product values between any two vectors.

Answer

Same magnitude as the original vectors.

Question 42

In the Gram-Schmidt process, what mathematical operation is used to ensure orthogonality between the generated vectors?

Accepted Answer

Projection of one vector onto another followed by subtraction.

Answer

Taking the cross product of the vectors.

Answer

Addition of scalar multiples of one vector to another.

Question 43

If the Gram-Schmidt process is applied to a set of linearly dependent vectors, what will be the outcome?

Accepted Answer

One or more of the resulting vectors will be the zero vector.

Answer

The process will fail to terminate.

Answer

The process will produce an orthonormal basis for the entire vector space.

Question 44

What is the relationship between an orthonormal basis and the Gram-Schmidt process?

Accepted Answer

An orthonormal basis is a set of orthogonal vectors with unit length, which can be obtained by applying the Gram-Schmidt process.

Answer

The Gram-Schmidt process can only be applied to orthonormal bases.

Answer

Orthonormal bases are orthogonal but not necessarily unit length.

Question 45

How is the Gram-Schmidt process related to the QR factorization of a matrix?

Accepted Answer

The Q matrix in the QR factorization contains the orthonormal vectors obtained from applying the Gram-Schmidt process to the columns of the original matrix.

Answer

The R matrix in the QR factorization contains the orthonormal vectors obtained from the Gram-Schmidt process.

Answer

The Gram-Schmidt process is a necessary step for performing QR factorization.

Question 46

In what applications is the Gram-Schmidt process commonly utilized?

Accepted Answer

Linear algebra, computer graphics, data science, and machine learning.

Answer

Statistics, probability, and combinatorial mathematics.

Answer

Number theory, abstract algebra, and topology.

Answer

Calculus, differential equations, and real analysis.

Question 47

What is a potential advantage of using an orthonormal basis obtained through the Gram-Schmidt process for solving systems of linear equations?

Accepted Answer

It can improve numerical stability and simplify calculations, leading to more accurate solutions.

Answer

It guarantees a unique solution to any system of linear equations.

Answer

It eliminates the need for back-substitution.

Question 48

Which of the following is a viable alternative to the Gram-Schmidt process for orthogonalizing a set of vectors?

Accepted Answer

Householder transformations

Answer

Gaussian elimination

Answer

Eigenvalue decomposition

Answer

Cramer's rule

Question 49

Which of the following is not a step in the Gram-Schmidt process?

Accepted Answer

Subtract the projection of the current vector onto the previous vectors

Answer

Project the vector onto the subspace spanned by the previous vectors

Answer

Normalize the vector

Question 50

What is the result of applying the Gram-Schmidt process to a set of linearly independent vectors?

Accepted Answer

An orthonormal basis for the subspace spanned by the original vectors

Answer

A set of linearly dependent vectors

Answer

A set of orthogonal vectors that are not normalized

Question 51

Which of the following is a property of an orthonormal basis?

Accepted Answer

The vectors are orthogonal to each other and have unit length

Answer

The vectors are linearly independent and span the subspace

Answer

The vectors are parallel to each other

Question 52

How can the Gram-Schmidt process be used to find the projection of a vector onto a subspace?

Accepted Answer

Project the vector onto each of the orthogonal vectors in the subspace and sum the projections

Answer

Subtract the vector from each of the orthogonal vectors in the subspace

Answer

Add the vector to each of the orthogonal vectors in the subspace

Question 53

Which of the following is an application of the Gram-Schmidt process in linear algebra?

Accepted Answer

Finding eigenvalues and eigenvectors of a matrix

Answer

Computing the determinant of a matrix

Answer

Inverting a matrix

Answer

Solving systems of differential equations

Question 54

What is the dimension of the subspace spanned by a set of n linearly independent vectors after applying the Gram-Schmidt process?

Accepted Answer

n

Answer

0

Answer

n - 1

Answer

n + 1

Question 55

Which of the following matrices is an orthogonal matrix?

Accepted Answer

A matrix whose columns form an orthonormal basis

Answer

A matrix whose rows form an orthonormal basis

Answer

A matrix whose inverse is equal to its transpose

Answer

A matrix whose determinant is 1

Question 56

What is the purpose of normalizing the vectors in the Gram-Schmidt process?

Accepted Answer

To make the vectors have unit length

Answer

To make the vectors orthogonal to each other

Answer

To make the vectors easier to compute

Answer

To make the vectors linearly independent

Question 57

Which of the following is a potential limitation of the Gram-Schmidt process?

Accepted Answer

It can be numerically unstable for ill-conditioned matrices

Answer

It is computationally expensive for large matrices

Answer

It can only be applied to matrices with real entries

Question 58

What is the relationship between the Gram-Schmidt process and QR factorization?

Accepted Answer

The Gram-Schmidt process can be used to compute the Q matrix in QR factorization

Answer

The Gram-Schmidt process is equivalent to QR factorization

Answer

The Q matrix in QR factorization is always an orthogonal matrix