Understanding the Gram-Schmidt Process

This video tutorial explains how to use the Gram-Schmidt process to compute an orthogonal basis for R3, defined by the span of three vectors. It begins by defining the vectors v1, v2, and v3, and then applies the Gram-Schmidt process to find the orthogonal basis vectors u1, u2, and u3. The tutorial includes detailed calculations for each step, ensuring the vectors are orthogonal by verifying their dot products. The video concludes with a confirmation of the orthogonality of the resulting vectors.

What is the main goal of using the Gram-Schmidt process in this example?

• To determine the inverse of a matrix

• To solve a system of linear equations

• To find a linearly dependent set of vectors

• To compute an orthogonal basis for R3

Which vector is directly used as the first orthogonal vector u1?

• Vector v3

• Vector v2

• Vector u2

• Vector v1

How is the second orthogonal vector u2 calculated?

• By adding vector v2 and vector u1

• By subtracting a scalar multiple of vector u1 from vector v2

• By multiplying vector v2 by a constant

• By dividing vector v2 by vector u1

What is the result of the dot product of vector v2 and vector u1?

• 1

• 2

• 3

• 0

What is the final form of the second orthogonal vector u2?

• Vector 1 1 0

• Vector 0 0 1

• Vector 1 1 1

• Vector 3 1 1

Which operation is crucial in determining the third orthogonal vector u3?

• Matrix inversion

• Dot product calculation

• Matrix multiplication

• Cross product calculation

What is the scalar used for vector u1 when determining vector u3?

• Different from the scalar for vector u2

• Zero

• One

• The same as for vector u2

What is the final form of the third orthogonal vector u3?

• Vector 1 1 0

• Vector 0 0 1

• Vector 3 1 1

• Vector 1 -1 0

What is verified by checking the dot products of the orthogonal vectors?

• The vectors are orthogonal

• The vectors are linearly dependent

• The vectors are identical

• The vectors are parallel

What is the significance of the orthogonal basis found in this example?

• It has the same span as the original vectors but is orthogonal

• It is not useful for any applications

• It is a subset of the original vectors

• It has a different span than the original vectors