Finding Coordinates Using Transition Matrices

The video tutorial explains how to find the coordinates of a vector relative to the standard basis using two methods. The first method involves direct calculation using the given basis vectors and coordinates. The second method uses a transition matrix to achieve the same result. Both methods yield the coordinates of vector w as (-11, 6, -25) relative to the standard basis.

What is the main task described in the introduction of the video?

• To find the coordinates of vector w relative to the standard basis.

• To calculate the length of vector w.

• To determine if the vectors in set B form a basis.

• To find the coordinates of vector w relative to a new basis.

In the first method, what is the first step to find the coordinates of vector w relative to the standard basis?

• Add the coordinates of vector w.

• Multiply the coordinates of vector w by the standard basis vectors.

• Multiply each coordinate of vector w by the corresponding basis vector in set B.

• Subtract the coordinates of vector w from the basis vectors.

What is the result of the first coordinate calculation using the first method?

• -25

• -11

• 0

• 6

What is the second method introduced to find the coordinates of vector w?

• Using a transition matrix from basis B to the standard basis.

• Using a different set of basis vectors.

• Using a graphical method.

• Using a calculator.

How is the transition matrix from basis B to the standard basis formed?

• By adding the basis vectors in set B.

• By using the standard basis vectors.

• By arranging the basis vectors in set B as columns.

• By subtracting the basis vectors in set B.

What is the first column of the transition matrix from basis B to the standard basis?

• 0, 0, 1

• -3, 2, -7

• -2, 1, -4

• 1, 4, -2

What is the second column of the transition matrix from basis B to the standard basis?

• -2, 1, -4

• 1, 4, -2

• -3, 2, -7

• 0, 0, 1

What is the third column of the transition matrix from basis B to the standard basis?

• -2, 1, -4

• 0, 0, 1

• 1, 4, -2

• -3, 2, -7

What is the result of the second coordinate calculation using the transition matrix method?

• -25

• 6

• 0

• -11

What is the result of the third coordinate calculation using the transition matrix method?

• 0

• 6

• -25

• -11