Understanding Vector Coordinates and Transition Matrices

The video tutorial explains how to find the coordinates of vector W relative to a new basis B, given its coordinates in the standard basis. It covers the concept of transition matrices, their inverses, and how to use them to transform vector coordinates. The process involves calculating the inverse of a 3x3 matrix formed by basis vectors and using it to find the new coordinates. The tutorial concludes with the final coordinates of vector W in basis B.

What is the initial task given in the problem regarding vector W?

• To find the length of vector W

• To find the angle between vector W and another vector

• To find the coordinates of vector W relative to basis B

• To find the magnitude of vector W

How is the transition matrix from S to B related to the transition matrix from B to S?

• It is the transpose of the transition matrix from B to S

• It is the inverse of the transition matrix from B to S

• It is the determinant of the transition matrix from B to S

• It is the same as the transition matrix from B to S

What is the first step in forming the matrix using basis vectors?

• Finding the determinant of the matrix

• Arranging the basis vectors as columns

• Calculating the eigenvalues

• Performing row reduction

What are the coordinates of vector W relative to the standard basis?

• 12, -4, 15

• 12, 4, 15

• -12, 4, -15

• -12, -4, 15

What is the first row of the inverse matrix used in the calculation?

• 1, -1, -1

• 0, 3, 1

• 0, 1, 0

• 1, 0, 0

What is the purpose of finding the inverse of the matrix formed using basis vectors?

• To calculate the determinant

• To find the transition matrix from S to B

• To perform a matrix transformation

• To solve a system of equations

What is the result of multiplying the first coordinate in the calculation?

• -3

• 4

• -1

• 0

What is the second coordinate of vector W relative to basis B?

• -1

• -3

• 4

• 0

What is the third coordinate of vector W relative to basis B?

• 0

• -3

• 4

• -1

What is the final conclusion about the coordinates of vector W relative to basis B?

• They are the same as the standard basis coordinates

• They are negative one, four, negative three

• They are all positive

• They are zero