Understanding the Gram-Schmidt Process

The video tutorial explains the Gram-Schmidt process for generating an orthonormal basis. It begins by defining a subspace V in R3 and finding a basis for it. The tutorial then demonstrates how to find an orthonormal basis by normalizing vectors and using projections to ensure orthogonality. The process is illustrated with examples, showing how to replace vectors with orthogonal ones and adjust their lengths to form an orthonormal basis. The video concludes with a complete orthonormal basis for the given subspace.

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

• To find a linearly dependent set of vectors

• To calculate the determinant of a matrix

• To generate an orthonormal basis for a subspace

• To solve a system of linear equations

How is the subspace V defined in terms of the plane equation x1 + x2 + x3 = 0?

• As the set of all vectors in R4

• As the set of all vectors in R3 that satisfy x1 - x2 - x3 = 0

• As the set of all vectors in R3 that satisfy x1 + x2 + x3 = 0

• As the set of all vectors in R2

What is the first step in finding an orthonormal basis for the span of v1?

• Adding a new vector to v1

• Dividing v1 by its length to create a unit vector

• Subtracting v2 from v1

• Multiplying v1 by a scalar

What is the length of the vector v1 before normalization?

• 1

• Square root of 2

• 2

• Square root of 3

What is the relationship between the vectors u1 and v1?

• u1 is a scaled version of v1

• u1 is perpendicular to v1

• u1 is the sum of v1 and v2

• u1 is unrelated to v1

What is the purpose of finding the vector y2 in the Gram-Schmidt process?

• To find a vector that is a linear combination of v1 and v2

• To find a vector that is a scalar multiple of v2

• To find a vector parallel to u1

• To find a vector orthogonal to u1

How is the vector y2 calculated in relation to v2 and its projection?

• y2 is the quotient of v2 and its projection onto v1

• y2 is the sum of v2 and its projection onto v1

• y2 is the difference between v2 and its projection onto v1

• y2 is the product of v2 and its projection onto v1

What is the final step to ensure that the vectors u1 and u2 form an orthonormal basis?

• Ensuring both vectors are parallel

• Ensuring both vectors have length 1

• Ensuring both vectors are linearly dependent

• Ensuring both vectors are identical

What is the significance of the vector u2 in the final orthonormal basis?

• u2 is parallel to u1

• u2 is a zero vector

• u2 is a duplicate of u1

• u2 is orthogonal to u1 and has length 1

What does the final orthonormal basis for the plane consist of?

• One vector that spans the plane

• Three vectors that are orthogonal

• Two vectors that are orthogonal and have length 1

• Two vectors that are linearly dependent