MMiDS 2.3: Self-Assessment Quiz

Let \(\mathbf{q}_1, \dots, \mathbf{q}_m\) be an orthonormal list of vectors in \(\mathbb{R}^n\). Which of the following is the orthogonal projection of a vector \(\mathbf{v} \in \mathbb{R}^n\) onto \(\mathrm{span}(\mathbf{q}_1, \dots, \mathbf{q}_m)\)?

• a) \(\sum_{i=1}^m \mathbf{q}_i\)
• b) \(\sum_{i=1}^m \langle \mathbf{v}, \mathbf{q}_i \rangle\)
• c) \(\sum_{i=1}^m \langle \mathbf{v}, \mathbf{q}_i \rangle \mathbf{q}_i\)
• d) \(\sum_{i=1}^m \langle \mathbf{q}_i, \mathbf{q}_i \rangle \mathbf{v}\)

What is the relationship between the dimensions of a linear subspace \(U\) of \(\mathbb{R}^n\) and its orthogonal complement \(U^\perp\)?

• a) \(\dim(U) = \dim(U^\perp)\)
• b) \(\dim(U) + \dim(U^\perp) = n\)
• c) \(\dim(U) \cdot \dim(U^\perp) = n\)
• d) There is no general relationship between \(\dim(U)\) and \(\dim(U^\perp)\).

For a given subspace \(U \subset \mathbb{R}^n\) and vector \(\mathbf{v} \in \mathbb{R}^n\), the orthogonal decomposition states that:

• a) \(\mathbf{v}\) can be written uniquely as the sum of a vector in \(U\) and a vector orthogonal to \(U\).
• b) \(\mathbf{v}\) can be written uniquely as a linear combination of vectors in \(U\).
• c) \(\mathbf{v}\) can be written uniquely as the product of a vector in \(U\) and a scalar.
• d) \(\mathbf{v}\) can be written uniquely as the product of two orthogonal vectors.

Let \(U\) be a linear subspace of \(\mathbb{R}^n\) and let \(\mathbf{v} \in \mathbb{R}^n\). Which of the following is the unique decomposition of \(\mathbf{v}\) guaranteed by the Orthogonal Decomposition Lemma?

• a) \(\mathbf{v} = \mathrm{proj}_U \mathbf{v} + (\mathrm{proj}_U \mathbf{v} - \mathbf{v})\)
• b) \(\mathbf{v} = \mathrm{proj}_U \mathbf{v} + (\mathbf{v} - \mathrm{proj}_{U^\perp} \mathbf{v})\)
• c) \(\mathbf{v} = \mathrm{proj}_U \mathbf{v} + (\mathbf{v} - \mathrm{proj}_U \mathbf{v})\)
• d) \(\mathbf{v} = \mathrm{proj}_{U^\perp} \mathbf{v} + (\mathbf{v} - \mathrm{proj}_{U^\perp} \mathbf{v})\)

According to the Normal Equations Theorem, what condition must a solution \(x^*\) to the linear least squares problem satisfy?

• a) \(A^T Ax^* = b\)
• b) \(A^T Ax^* = A^T b\)
• c) \(Ax^* = A^T b\)
• d) \(Ax^* = b\)

Under what condition is the solution to the linear least squares problem unique?

• a) When the rows of \(A\) are linearly independent.
• b) When the columns of \(A\) are linearly independent.
• c) When \(A\) is a square matrix.
• d) When \(A\) is an invertible matrix.

Which property characterizes the orthogonal projection \(\mathrm{proj}_U \mathbf{v}\) of a vector \(\mathbf{v}\) onto a subspace \(U\)?

• a) \(\mathbf{v} - \mathrm{proj}_U \mathbf{v}\) is a scalar multiple of \(\mathbf{v}\).
• b) \(\mathbf{v} - \mathrm{proj}_U \mathbf{v}\) is orthogonal to \(\mathbf{v}\).
• c) \(\mathbf{v} - \mathrm{proj}_U \mathbf{v}\) is orthogonal to \(U\).
• d) \(\mathrm{proj}_U \mathbf{v}\) is always the zero vector.

What is the interpretation of the linear least squares problem \(A\mathbf{x} \approx \mathbf{b}\) in terms of the column space of \(A\)?

• a) Finding the exact solution \(\mathbf{x}\) such that \(A\mathbf{x} = \mathbf{b}\).
• b) Finding the vector \(\mathbf{x}\) that makes the linear combination \(A\mathbf{x}\) of the columns of \(A\) as close as possible to \(\mathbf{b}\) in Euclidean norm.
• c) Finding the orthogonal projection of \(\mathbf{b}\) onto the column space of \(A\).
• d) Finding the orthogonal complement of the column space of \(A\).

Which matrix equation must hold true for a matrix \(Q\) to be orthogonal?

• a) \(Q Q^T = 0\).
• b) \(Q Q^T = I\).
• c) \(Q^T Q = 0\).
• d) \(Q^T = Q\).

Which of the following Python commands can be used to solve a linear system of equations in NumPy?

• a) `numpy.linalg.inv`
• b) `numpy.linalg.solve`
• c) `numpy.linalg.lstsq`
• d) `numpy.linalg.matrix_power`