If vector, v = (1,4) then find the matrix of perp v

Question

Can anyone please help me with this problem.

A vector $\boldsymbol{v}$ is given with the coordinates $1, 4$

$\boldsymbol{v} = (1,4)$

Find the matrix of perp $\boldsymbol{v}$.

As far as I know perp $\boldsymbol{v}$ means the orthogonal compliment of $\boldsymbol{v}$.

I dont know how to find perp v, but I know how to find the matrix if two vectors are given. Can anyone please help me out? Please be gentle, I'm very new to linear algebra ^-^

EDIT: I see a lot of people are asking if I stated the problem correctly.

Well, I have finally taken a picture of the question from my book. Hope this clears up all confusions and doubts

If it helps the answer is given as:

1/17 (16 -4) (-4 1) . Its a matrix, sorry if I couldnt write the mathematical symbols correctly.

If it helps to understand the matrix more clearly, 1st coloumn of the matrix is (16, -4) second coloumn is (-4, 1) and the matrix is multiplied by 1/17 (a scalar)

But thats the answer. Could anyone please help me guide through the steps, to get to this answer? What does perpv mean here? Is it the same as an orthogonal compliment? How do I find the orthogonal compliments matrix?

Answer 1

A perpendicular matrix is just changing the entries of the vector, so that

\begin{align*} A \cdot \vec{x} = A \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_2 \\ x_1 \end{pmatrix} \end{align*}

The perpendicular matrix is (I think in general):

\begin{align*} A= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{align*}

A projection matrix just projects a vector on the zero (or the kernel): \begin{align*} A \cdot \vec{x} = 0\end{align*}

Now just write down the equations, I think you will mention that.