Question

Consider the matrix \begin{bmatrix} 1 & -6 \\ 3 & 6 \\ 4 & 8 \\ 5 & 0 \\ 7 & 8 \end{bmatrix}. (a) Let $\beta = \{\vec{q}_1, \vec{q}_2, \vec{q}_3, \vec{q}_4, \vec{q}_5\}$ be an orthonormal basis for $\mathbb{R}^5$, such that $\vec{q}_1, \vec{q}_2$ span the column space of A. Which of the four fundamental subspaces is spanned by $\vec{q}_3, \vec{q}_4, \vec{q}_5$? (b) Find $\vec{q}_1$ and $\vec{q}_2$. (c) Find $\beta$. You should find $\vec{q}_3, \vec{q}_4$, and $\vec{q}_5$. (d) Find Q and R in the QR-decomposition of A. (e) Using your result in (4d), find the best solution to the equation $A\vec{x} = \vec{b}$, if $\vec{b} =$ $(-3, 7, 1, 0, 4)$.

          Consider the matrix
\begin{bmatrix} 1 & -6 \\ 3 & 6 \\ 4 & 8 \\ 5 & 0 \\ 7 & 8 \end{bmatrix}.
(a) Let $\beta = \{\vec{q}_1, \vec{q}_2, \vec{q}_3, \vec{q}_4, \vec{q}_5\}$ be an orthonormal basis for $\mathbb{R}^5$, such that $\vec{q}_1, \vec{q}_2$
span the column space of A. Which of the four fundamental subspaces is spanned
by $\vec{q}_3, \vec{q}_4, \vec{q}_5$?
(b) Find $\vec{q}_1$ and $\vec{q}_2$.
(c) Find $\beta$. You should find $\vec{q}_3, \vec{q}_4$, and $\vec{q}_5$.
(d) Find Q and R in the QR-decomposition of A.
(e) Using your result in (4d), find the best solution to the equation $A\vec{x} = \vec{b}$, if $\vec{b} =$
$(-3, 7, 1, 0, 4)$.

$Consider the matrix < b m a t r i x > . (a) Let β = {q⃗1, q⃗2, q⃗3, q⃗4, q⃗5} be an orthonormal basis for ℝ^5, such that q⃗1, q⃗2 span the column space of A. Which of the four fundamental subspaces is spanned by q⃗3, q⃗4, q⃗5? (b) Find q⃗1 and q⃗2. (c) Find β. You should find q⃗3, q⃗4, and q⃗5. (d) Find Q and R in the QR-decomposition of A. (e) Using your result in (4d), find the best solution to the equation Ax⃗ = b⃗, if b⃗ = (-3, 7, 1, 0, 4).$

Added by John W.

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