MMiDS 2.4: Self-Assessment Quiz

Which of the following matrices is upper triangular?

• a) \(\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\)
• b) \(\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}\)
• c) \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
• d) \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\)

What is the output of the Gram-Schmidt algorithm when applied to a set of linearly independent vectors?

• a) An orthogonal basis for the subspace spanned by the vectors.
• b) An orthonormal basis for the subspace spanned by the vectors.
• c) A set of linearly dependent vectors.
• d) A single vector that is orthogonal to all the input vectors.

In the QR decomposition \(A = QR\), what properties do the matrices \(Q\) and \(R\) have?

• a) \(Q\) is orthogonal, and \(R\) is lower triangular.
• b) \(Q\) is orthogonal, and \(R\) is upper triangular.
• c) \(Q\) is upper triangular, and \(R\) is orthogonal.
• d) \(Q\) is lower triangular, and \(R\) is orthogonal.

How can the QR decomposition be used to solve the linear least squares problem \(\min_{x \in \mathbb{R}^m} \|Ax - b\|\)?

• a) \(x^* = Q^Tb\)
• b) \(x^* = R^{-1}Q^Tb\)
• c) \(Rx^* = Q^Tb\)
• d) \(Qx^* = R^Tb\)

Which of the following statements about the QR decomposition is true?

• a) The reduced QR decomposition has an orthogonal \(Q\) matrix and an upper triangular \(R\) matrix.
• b) The full QR decomposition has an orthogonal \(Q\) matrix and a lower triangular \(R\) matrix.
• c) The reduced QR decomposition has a square orthogonal \(Q\) matrix and an upper triangular \(R\) matrix.
• d) The full QR decomposition has a square orthogonal \(Q\) matrix and an upper triangular \(R\) matrix.

Which step is NOT part of the back substitution process to solve \(Rx = b\)?

• a) Start with the last row of the system.
• b) Assume that \(r_{mm} \neq 0\).
• c) Use forward elimination to reduce the matrix.
• d) Solve for \(x_i\) by considering the known values of \(x_{i+1}, \ldots, x_m\).

What is the dimension of a hyperplane in \(\mathbb{R}^m\)?

• a) \(m\)
• b) \(m - 1\)
• c) \(m - 2\)
• d) 1

Which of the following is true about a Householder reflection?

• a) It is a reflection across a hyperplane orthogonal to a unit vector.
• b) It is a reflection across a unit vector.
• c) It is a rotation around a hyperplane orthogonal to a unit vector.
• d) It is a rotation around a unit vector.

Which of the following is true about a Householder reflection matrix \(H = I_{m \times m} - 2zz^T\)?

• a) \(H\) is symmetric.
• b) \(H\) is skew-symmetric.
• c) \(H\) is orthogonal.
• d) Both a and c.

How can a sequence of Householder reflections be used to compute a QR decomposition of a matrix \(A\)?

• a) By iteratively introducing zeros above the diagonal of \(A\).
• b) By iteratively introducing zeros below the diagonal of \(A\).
• c) By iteratively introducing zeros on the diagonal of \(A\).
• d) By iteratively introducing zeros in the entire matrix \(A\).

If \(H = I_{m \times m} - 2zz^T\) is a Householder reflection, what is the relationship between \(z\) and the hyperplane \(W\) being reflected across?

• a) \(z\) is a vector in \(W\).
• b) \(z\) is orthogonal to \(W\).
• c) \(z\) is parallel to \(W\).
• d) \(z\) is the normal vector to \(W\).

If \(Q\) is an orthogonal matrix, which of the following is true for any vectors \(x\) and \(y\)?

• a) \(\|Qx\| = \|x\|\) and \(\langle Qx, Qy \rangle = \langle x, y \rangle\)
• b) \(\|Qx\| \neq \|x\|\) but \(\langle Qx, Qy \rangle = \langle x, y \rangle\)
• c) \(\|Qx\| = \|x\|\) but \(\langle Qx, Qy \rangle \neq \langle x, y \rangle\)
• d) \(\|Qx\| \neq \|x\|\) and \(\langle Qx, Qy \rangle \neq \langle x, y \rangle\)