What is the rank of the matrix

$\left[ \begin{array}{rrr} 1 & 2 & 3 \\ 2 & 5 & 2 \\ 6 & -3 & 1 \end{array} \right]$

Let $X$ be an $m \times n$ matrix and let $H = X (X^{T} X)^{-1} X^{T}$ (assume that $X$ is full rank so that $(X^{T} X)^{-1}$ exists). Give an expression for $H^{T}$ in terms of the matrix $X$. Simplify your answer.

[Hint: write X^(T) for the transpose of X]

Let $X$ be an $m \times n$ matrix and let $H = X (X^{T} X)^{-1} X^{T}$ (assume that $X$ is full rank so that $(X^{T} X)^{-1}$ exists). Give an expression for $H^{2}$ in terms of the matrix $X$. Simplify your answer.

[Hint: write X^(T) for the transpose of X]

Classify the following vector pairs as orthonormal, orthogonal, or only independent.

$(a) \; \left[ \begin{array}{r} 1 \\ 0 \end{array} \right] \; \mbox{&} \, \left[ \begin{array}{r} -1 \\ 1 \end{array} \right] \quad (b) \; \left[ \begin{array}{r} .6 \\ .8 \end{array} \right] \; \mbox{&} \, \left[ \begin{array}{r} .4 \\ -.3 \end{array} \right] \quad (c) \; \left[ \begin{array}{r} \cos(\theta) \\ \sin(\theta) \end{array} \right] \; \mbox{&} \, \left[ \begin{array}{r} -\sin(\theta) \\ \cos(\theta) \end{array} \right]$

Answer the question by entering the letters a, b, and c (each once) separated by spaces. The order should be the orthonormal pair, the orthogonal pair, the independent pair.

Let $A = S \Lambda S^{-1}$ be the factorization of $A$ into its eigenvectors and eigenvalues where

$\Lambda = \left[ \begin{array}{cccc} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0& d \end{array} \right]$

What is the largest eigenvalue of $A^{7}$?

Use calculus to find the slope of the line through the origin (i.e., $y = mx$) that minimizes the squared distance to the points $(1, 1)$, $(2, 3)$, $(4, 2)$. What is the slope $m$ rounded to $2$ significant digits?