# Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$

## Problem 641

Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$.

Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.

## Solution.

Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$.

Then we want
$\mathbf{v} \mathbf{w} =\begin{bmatrix} 2 & -5 & -1 \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}=2 w_1 – 5 w_2 – w_3 = 0.$

Letting $w_1$ and $w_2$ be free variables, we solve $w_3 = 2w_1 – 5w_2$.
Then every solution to the equation $\mathbf{v} \mathbf{w} = 0$ is of the form
$\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ 2 w_1 – 5 w_2 \end{bmatrix} = w_1 \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} + w_2 \begin{bmatrix} 0 \\ 1 \\ -5 \end{bmatrix}.$

##### Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$
Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.