Linear Algebra Quiz: Best Approximation

Let $\mathbf{y} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and let $W$ be the subspace spanned by the orthogonal vectors $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$. What is the shortest distance from $\mathbf{y}$ to $W$?

Let $W$ be the subspace of $\mathbb{R}^3$ spanned by the orthogonal vectors $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$. Find the best approximation of $\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ in $W$.

Let $W$ be a subspace of $\mathbb{R}^n$. If a vector $\mathbf{v}$ is an element of $W$, what is the best approximation of $\mathbf{v}$ in $W$?

What point in the plane spanned by the orthogonal vectors $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ is closest to the point $P(3, 3, 3)$?

Let $W$ be a subspace with an orthonormal basis ${\mathbf{q}_1, \mathbf{q}_2}$, where $\mathbf{q}_1 = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{q}_2 = \frac{1}{\sqrt{6}}\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$. Find the best approximation of $\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ in $W$.

In a signal processing application, a target signal is represented by the vector $\mathbf{s} = \begin{pmatrix} 10 \\ 5 \\ 15 \end{pmatrix}$. This signal is to be approximated using a linear combination of two orthogonal basis signals, $\mathbf{b}_1 = \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix}$ and $\mathbf{b}_2 = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$. What is the resulting best-fit approximation of $\mathbf{s}$?

Let $W$ be a subspace of $\mathbb{R}^n$, and let $\mathbf{v}$ be a vector in $\mathbb{R}^n$. Let $\hat{\mathbf{w}} = \mathrm{proj}_W(\mathbf{v})$ be the best approximation of $\mathbf{v}$ in $W$. What is the best approximation of the vector $\mathbf{z} = \mathbf{v} - \hat{\mathbf{w}}$ in the same subspace $W$?

Let $W$ be the subspace of $\mathbb{R}^3$ spanned by the orthogonal basis ${\mathbf{u}_1 = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}, \mathbf{u}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}}$. For the vector $\mathbf{v} = \begin{pmatrix} 3 \\ 0 \\ 6 \end{pmatrix}$, is the vector $\mathbf{w}_c = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$ the best approximation of $\mathbf{v}$ in $W$?

Let $\mathbb{R}^n$ be equipped with the standard dot product. Let $W$ be a subspace of $\mathbb{R}^n$ and let $\mathbf{v}$ be a vector in $\mathbb{R}^n$. A vector $\hat{\mathbf{w}}$ is the best approximation to $\mathbf{v}$ in $W$. Which statement provides the unique defining property of $\hat{\mathbf{w}}$?

Let $W$ be the plane in $\mathbb{R}^3$ spanned by $\mathbf{x}_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{x}_2 = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$. Which vector in $W$ is closest to $\mathbf{v} = \begin{pmatrix} 0 \\ 3 \\ 3 \end{pmatrix}$?

In the inner product space $C[0,1]$ with inner product $\langle f, g \rangle = \int_0^1 f(x)g(x)dx$, let $W = \text{span}\{1, x\}$ and $f(x) = x^2$. The best approximation to $f$ in $W$ has the form $ax + b$. What is the value of $a$?

Let $\mathbf{v}_1 = (1, 2, 1)$, $\mathbf{v}_2 = (0, 1, 2)$, and $\mathbf{v}_3 = (2, 3, 0)$ in $\mathbb{R}^3$. If $W = \text{span}\{\mathbf{v}_1, \mathbf{v}_2\}$ and $\mathbf{b} = (3, 4, 5)$, which statement about the best approximation $\hat{\mathbf{b}}$ to $\mathbf{b}$ in $W$ is correct?

Consider the matrix $A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ 1 & 2 \end{pmatrix}$ and vector $\mathbf{b} = \begin{pmatrix} 3 \\ 7 \\ 2 \end{pmatrix}$. The least squares solution $\hat{\mathbf{x}}$ to $A\mathbf{x} = \mathbf{b}$ minimizes $||A\mathbf{x} - \mathbf{b}||^2$. What is the minimum value of this expression?

Let $A$ be an $m \times n$ matrix with $\text{rank}(A) = r < n$, and consider the least squares problem $\min_{\mathbf{x}} ||A\mathbf{x} - \mathbf{b}||^2$. If $\mathbf{x}_0$ is any solution to the normal equations $A^TA\mathbf{x} = A^T\mathbf{b}$, which vector represents the best approximation to $\mathbf{b}$ in the column space of $A$?

Let $S = \{\mathbf{u}_1, \mathbf{u}_2\}$ be an orthonormal set in an inner product space $V$, and let $W = \text{span}(S)$. If $\mathbf{v} \in V$ and $\langle \mathbf{v}, \mathbf{u}_1 \rangle = 4$ and $\langle \mathbf{v}, \mathbf{u}_2 \rangle = -3$, what is $||\mathbf{v} - \text{proj}_W(\mathbf{v})||^2$?

In $\mathbb{R}^3$, let $L$ be the line through the origin with direction vector $\mathbf{d} = (2, -1, 2)$. If $\mathbf{p} = (6, 3, 0)$ and $\mathbf{q}$ is the best approximation to $\mathbf{p}$ on $L$, what is $|\mathbf{p} - \mathbf{q}|^2 + |\mathbf{q}|^2$?

Find the best approximation of the vector $\mathbf{v} = \begin{pmatrix} 7 \\ 6 \end{pmatrix}$ by a vector on the line spanned by $\mathbf{u} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$.