Rabbits Nine

Is there a transformation done with a matrix that is addition?

M\begin{bmatrix} v^1 \\ v^2 \end{bmatrix} = \begin{bmatrix} v^{1'} \\ v^{2'} \end{bmatrix}

There is an answer to the question in Appendix A.

A vector isn’t tied to a coordinate system. We can move the point at (1,3) to (5,8).

(1,3) + (4,5) = (5,8)

But a point is not a vector. One interpretation says that a vector takes a point in an old location and moves it to a new location.

Appendix A

If we want to add (a,b) to (x,y) we need to write the vector as \begin{bmatrix} x\\y\\1 \end{bmatrix} and then we can do following matrix multiplication:

\begin{bmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x\\y\\1 \end{bmatrix} = \begin{bmatrix} x+a\\y+b\\1 \end{bmatrix}

If we need this when using the warp drive of our spaceship, I’ll use it. However, it is quite bothersome that we need three components to represent a vector in a two dimensional space.

Perhaps this discussion is valuable fir making a vector look like \begin{bmatrix} \Delta x_1 \\ \Delta x_2 \\ \Delta x_3 \end{bmatrix} .

There is another issue. A vector is independent of a coordinate system. We might draw a vector from (0,0) to (1,1) and then elsewhere draw a vector from (2,5) to (3,6) and the graph paper says that we moved from (1,1) to (3,6).

In contradiction, both vectors say “I am the vector (1,1).”