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) $\sum _{i=1}^m⟨\mathbf{v},{\mathbf{q}}_i⟩$
• c) $\sum _{i=1}^m⟨\mathbf{v},{\mathbf{q}}_i⟩{\mathbf{q}}_i$
• d) $\sum _{i=1}^m⟨{\mathbf{q}}_i,{\mathbf{q}}_i⟩\mathbf{v}$

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 $x^{\ast }$ to the linear least squares problem satisfy?

• a) $A^{T}A{x}^{\ast }=b$
• b) $A^{T}A{x}^{\ast }=A^{T}b$
• c) $A{x}^{\ast }=A^{T}b$
• d) $A{x}^{\ast }=b$

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`