Unit 6 test: Orthogonality and Least Squares

Compute $(3, -2, 5) \cdot (1, 4, 2)$.

Find the distance between $(1, 0, -2)$ and $(5, -1, 6)$.

Find $k$ so that $(1, k, 3)$ and $(k, 2, -4)$ are orthogonal.

Let $W = \operatorname{Span}\{(1, 1, 1), (1, 2, 3)\}$. Find the vector in $W^\perp$ whose third entry is $1$.

$A$ is a $4 \times 6$ matrix. Which subspace is the orthogonal complement of $\operatorname{Row} A$?

The vectors $\mathbf{u}_1 = (1, 1, 1)$, $\mathbf{u}_2 = (1, -2, 1)$ and $\mathbf{u}_3 = (1, 0, -1)$ form an orthogonal basis for $\mathbb{R}^3$. Find the weights $(c_1, c_2, c_3)$ with $(2, -3, 4) = c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + c_3\mathbf{u}_3$.

$U$ is an $n \times n$ orthogonal matrix. Which statement must be true?

Find the orthogonal projection of $\mathbf{y} = (3, 4)$ onto the line spanned by $\mathbf{u} = (2, 1)$.

Let $\mathbf{u}_1 = (1, 1, 1, 1)$, $\mathbf{u}_2 = (1, 1, -1, -1)$ and $\mathbf{y} = (5, 1, 2, 0)$. Find the orthogonal projection of $\mathbf{y}$ onto $W = \operatorname{Span}\{\mathbf{u}_1, \mathbf{u}_2\}$.

Find the distance from $\mathbf{y} = (3, 5, 1)$ to the plane $W = \operatorname{Span}\{(1, 0, 1), (0, 1, 0)\}$. Give an exact answer.

Apply the Gram-Schmidt process to $\mathbf{x}_1 = (1, 0, 1)$ and $\mathbf{x}_2 = (3, 2, 1)$. Find $\mathbf{v}_2$.

Let $A = \begin{bmatrix} 2 & 6 \\ 2 & 0 \\ 1 & 6 \end{bmatrix}$. In the QR factorization $A = QR$, what is $R$?

Find the least-squares solution of $A\mathbf{x} = \mathbf{b}$ for $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}$ and $\mathbf{b} = (1, 2, 4)$.

Find the slope of the least-squares line for the points $(0, 3)$, $(1, 2)$, $(2, 0)$, $(3, -1)$.

Find the least-squares error $\|\mathbf{b} - A\hat{\mathbf{x}}\|$ for $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{bmatrix}$ and $\mathbf{b} = (0, 0, 6)$. Give an exact answer.