Understanding B Coordinates in Orthogonal and Orthonormal Bases

This video tutorial explains how to find the B coordinates of a vector when B is an orthogonal set. It begins with an introduction to the concept and provides a step-by-step example. The tutorial checks the orthogonality of the basis vectors and calculates the B coordinates using a specific formula. It concludes with a verification of the calculations and a graphical representation of the vectors involved.

What is the primary goal of this lesson?

• To understand the concept of vector addition.

• To study the history of vector spaces.

• To learn how to find the B coordinates of a vector in an orthogonal set.

• To explore the properties of matrices.

What is the key difference between orthogonal and orthonormal bases?

• Orthonormal bases are always in three dimensions.

• Orthogonal bases are always in two dimensions.

• Orthonormal bases have unit vectors, while orthogonal bases do not.

• Orthogonal bases have unit vectors, while orthonormal bases do not.

What is the first step in solving the example problem?

• Calculating the magnitude of vector x.

• Checking if the vectors in B are orthogonal.

• Finding the inverse of the matrix.

• Determining if vector x is a unit vector.

What is the significance of the dot product being zero in the example?

• It indicates that the vectors are parallel.

• It confirms that the vectors are orthogonal.

• It shows that the vectors are identical.

• It means the vectors are in different dimensions.

How do you calculate the first component of the B coordinates?

• By subtracting vector u2 from vector x.

• By dividing the dot product of vector x and vector u1 by the dot product of vector u1 with itself.

• By adding the vectors u1 and u2.

• By multiplying the vectors u1 and u2.

What are the calculated B coordinates of vector x in the example?

• One-half and one-third

• Six-fifths and three-tenths

• Five-sixths and one-fifth

• Two-thirds and one-fourth

Why is the basis in the example not orthonormal?

• Because the vectors are not parallel.

• Because the vectors are not in the same plane.

• Because the vectors are not unit vectors.

• Because the vectors are not orthogonal.

What is the result of the dot product of vector u1 with itself in the example?

• Two

• Three

• Four

• Five

What does the graphical representation confirm about the B coordinates?

• That vector x is not part of the subspace.

• That the calculated B coordinates correctly represent vector x.

• That the vectors are not orthogonal.

• That the calculated B coordinates are incorrect.

What is the final check performed in the example?

• Verifying the orthogonality of the vectors.

• Checking the graphical representation of the vectors.

• Calculating the inverse of the matrix.

• Finding the determinant of the vectors.