Understanding Fundamental Subspaces through SVD

This video tutorial explains how to determine the fundamental subspaces of a matrix using its singular value decomposition (SVD). It covers the null space, column space, and row space of a matrix and its transpose. The tutorial demonstrates how to use the matrices U, Sigma, and V transpose from the SVD to find orthonormal bases for these subspaces. The video also provides a step-by-step guide to identifying these subspaces and concludes with a summary of the findings.

What are the fundamental subspaces of a matrix A?

• Eigen space of A, eigen space of A transpose, null space of A, null space of A transpose

• Null space of A, null space of A transpose, column space of A, column space of A transpose

• Row space of A, row space of A transpose, null space of A, null space of A transpose

• Diagonal space of A, diagonal space of A transpose, column space of A, column space of A transpose

What is the significance of the matrix U in the SVD of matrix A?

• It provides an orthonormal basis for the column space of A

• It provides an orthonormal basis for the null space of A

• It provides an orthonormal basis for the row space of A

• It provides an orthonormal basis for the diagonal space of A

What is the role of the matrix Sigma in the SVD of matrix A?

• It provides the orthonormal basis for the null space of A

• It provides the orthonormal basis for the column space of A

• It contains the eigenvalues of A

• It contains the singular values of A

How do you determine the column space of matrix A using SVD?

• By using the rows of matrix V transpose

• By using the columns of matrix V

• By using the diagonal elements of matrix Sigma

• By using the columns of matrix U

What does it mean for vectors to be orthonormal?

• They are unit vectors and identical

• They are orthogonal and parallel

• They are unit vectors and parallel

• They are unit vectors and orthogonal

What happens if matrix U has fewer columns than expected when finding the null space of A transpose?

• The null space is spanned by the identity matrix

• The null space is spanned by the columns of V

• The null space is spanned by the zero vector

• The null space is spanned by the diagonal elements of Sigma

How can the null space of A transpose be described if the necessary columns in matrix U are missing?

• As the span of the identity matrix

• As the span of the columns of V

• As the span of the zero vector

• As the span of the diagonal elements of Sigma

Which vectors are used to find the column space of A transpose?

• The columns of matrix V

• The columns of matrix U

• The diagonal elements of matrix Sigma

• The rows of matrix V transpose

What is the relationship between the column space of A transpose and the row space of A?

• They are equal

• They are orthogonal

• They are perpendicular

• They are independent

How is the null space of A determined using SVD?

• By using the first few rows of matrix U

• By using the first few columns of matrix V

• By using the last few rows of matrix V transpose

• By using the last few columns of matrix U