Find Matrix From Given Kernel

Question

I have been stuck on this problem for days now and any help is appreciated.

Express the $\mathrm{span}(E)$, which is the the two vectors below, as a kernel of a $3×3$ matrix:

\begin{pmatrix} 2\\3\\-5\ \end {pmatrix}, \begin{pmatrix} -4 \\-5 \\ 8 \end{pmatrix}

Answer 1

Since the rank of the matrix is 1, it suffices to find a row vector $r=(a,b,c)$ such that

• $r\cdot v_1=0$

• $r\cdot v_2=0$

and use this row vector for all the rows of the matrix.

Note that we can simplify the derivation by combining the vectors, for example as follow

$$v_3=2v_1+v_2=\begin{pmatrix} 0\\1\\-2\ \end {pmatrix}$$

$$v_4=v_1-3v_3=\begin{pmatrix} 2\\0\\1\ \end {pmatrix}$$

and then use $v_3$ and $v_4$ which allow to find directly a solution that is

$$\begin{pmatrix} 1&-4&-2\\1&-4&-2\\1&-4&-2\ \end {pmatrix}$$

Answer 2

Hints . . .

• Let $u = \langle{2,3,-5}\rangle$, and let $v=\langle{-4,-5,-8}\rangle$.$\\[4pt]$
• Are the vectors $u,v$ linearly independent?$\\[4pt]$
• What is the dimension of the span of $\{u,v\}$?$\\[4pt]$
• Let $w$ be a nonzero vector such that $w$ is orthogonal to both $u$ and $v$. For example, you can take $w=u{\,\times\,}v$.$\\[4pt]$
• Let $A$ be the $3{\,\times\,}3$ matrix with all rows equal to $w$.$\\[4pt]$
• What can you say about $Au$?$\\[4pt]$
• What can you say about $Av$?$\\[4pt]$
• For $a,b\in\mathbb{R}$, what can you say about $A(au+bv)$?$\\[4pt]$
• What is the rank of $A$?$\\[4pt]$
• What is the dimension of the kernel of $A$?$\\[4pt]$
• How does the kernel of $A$ relate to the span of $\{u,v\}$?