Orthonormal Basis and Gram-Schmidt Process

The video tutorial demonstrates the Gram-Schmidt process to find an orthonormal basis for a subspace in R4. It begins by introducing the vectors spanning the subspace and proceeds to normalize and orthogonalize each vector step-by-step. The process involves calculating projections and normalizing vectors to form an orthonormal basis. The tutorial concludes with a complete orthonormal basis for the subspace.

What is the main goal of the Gram-Schmidt process as introduced in the video?

• To find a set of linearly dependent vectors

• To create an orthonormal basis for a subspace

• To calculate the determinant of a matrix

• To solve a system of linear equations

Which vector is chosen first for normalization in the Gram-Schmidt process?

• u1

• v1

• v3

• v2

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What is the length of the vector v1 before normalization?

• 2

• 1

• Square root of 3

• Square root of 2

How is the vector y2 obtained from v2?

• By adding v2 to u1

• By dividing v2 by its length

• By multiplying v2 by a scalar

• By subtracting the projection of v2 onto u1 from v2

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What is the length of the vector y2 before it is normalized?

• Square root of 2

• Square root of 3/2

• 1

• 2

What is the purpose of scaling the vector y3 in the process?

• To make it orthogonal to u1 and u2

• To simplify the normalization process

• To change its direction

• To increase its length

What is the final step in creating an orthonormal basis for the subspace V?

• Normalizing the vector y3

• Finding the determinant of the matrix

• Adding all vectors together

• Projecting v3 onto u1 and u2

Which vectors form the orthonormal basis for the subspace V?

• u1, u2, u3

• y1, y2, y3

• u1, u2, v3

• v1, v2, v3

What is the significance of having an orthonormal basis?

• It reduces the number of vectors needed

• It makes the vectors linearly dependent

• It increases the dimension of the subspace

• It simplifies the computation of projections

What is the relationship between the original vectors and the orthonormal basis vectors?

• They are identical

• The orthonormal basis vectors are scaled versions of the original vectors

• The original vectors are orthogonal and have unit length

• The orthonormal basis vectors are orthogonal and have unit length