Vector Projection and Distance in R4

This video tutorial explains how to find the projection of a vector onto a subspace in R4 with an orthogonal basis. It provides a step-by-step example, calculating the orthogonal projection of a given vector onto a subspace and determining the distance from the vector to the subspace. The tutorial verifies the orthogonality of the basis vectors and uses dot products to compute the projection. It concludes by finding the projection onto the orthogonal complement and calculating the distance using vector magnitudes.

What is the main focus of the video tutorial?

• Finding the projection of a vector onto a subspace in R4

• Solving linear equations

• Calculating the determinant of a matrix

• Understanding eigenvalues and eigenvectors

What is the formula used for in the context of the video?

• Calculating the inverse of a matrix

• Finding the orthogonal projection of a vector onto a subspace

• Solving quadratic equations

• Determining the eigenvalues of a matrix

What is the vector X in the example provided?

• (1, 0, 0, 1)

• (0, 1, 3, 4)

• (2, 2, 2, 2)

• (1, 1, 1, 1)

What is the first step in verifying the orthogonality of the basis vectors?

• Finding the angle between the vectors

• Checking the dot products of the vectors

• Calculating the cross product of the vectors

• Checking the magnitude of each vector

What is the result of the dot product of vector X and vector U1?

• 1

• 0

• -3

• 3

What is the orthogonal projection of vector X onto W?

• (0, 0, 0, 0)

• (1, 1, 1, 1)

• (1/2, 1/2, 7/2, 7/2)

• (2, 2, 2, 2)

How is the projection of vector X onto the orthogonal complement of W found?

• By dividing vector X by vector Xw

• By multiplying vector X by a scalar

• By subtracting vector Xw from vector X

• By adding vector X and vector Xw

What is the magnitude of the vector representing the distance from X to W?

• 0

• 1

• 3

• 2

What does the distance from vector X to W represent?

• The angle between X and W

• The shortest distance from X to the subspace W

• The sum of the components of X

• The length of vector X

What is the final conclusion about the distance from vector X to W?

• It is three units

• It is two units

• It is one unit

• It is zero