# Solving system of equations for coordinates

This is my first time here, so let me know if there's something I need to add to the post etc. Anyway, I need some help with Mathematica and linear algebra, and I got some tips that this would be the right place to come for help.

I've got two bases, and a vector called x.

(V) First Base: {{1,3}, {4,6}}
(W) Second Base: {{4,6}, {2,5}}
Vector x: {6,6}


I started out by plotting the two base-vectors in the V-base and the vector x in the same picture. I then did the same thing with the W-base and the x vector.

I've plotted two graps, but I'll only post a picture of the first one initially. This is what the first plot looked like (for reference):

Now, here's what I need help with. I'm supposed to solve the system of equations in Mathematica, to find V-coordinates for x and W-coordinates for x, and then plot the components for x in each coordinate-system, using (Dashed, Line).

This graph was posted as an example on how it should look like.

• What have you tried so far? Have a look at LinearSolve and related functions. – Lukas Lang Jan 2 at 15:47
• Honestly, I haven't been able to do anything with it. I've been sitting all day with different exercises, and I haven't gotten anywhere with this one. That's why I figured I needed a good explanation.. – D.John Jan 2 at 16:04

Here is one solution:

V = {{1, 3}, {4, 6}};

x = {6, 6};

vx = LinearSolve[Transpose@V, x]
(* { -2, 2 } *)

vector[x0_: {0, 0}, v_] := Arrow@{x0, x0 + v}

Graphics[
{
Blue,
vector /@ V,
Red,
vector@x,
Black,
Dashed,
vector /@ (vx V),

• Why should it not go to negative coordinates? And yes, replacing all occurrences of V with W will do the same for the W base. But I'd suggest you try to understand what the code does in detail instead of just applying it - if you have questions about a specific line, feel free to ask – Lukas Lang Jan 2 at 16:41
• The vector function is a helper function that returns an Arrow directive that represents a vector starting at x0 and being v long (i.e. it ends at x0+v). The :{0,0} just means that if we only give one parameter to vector, x0 should take a default value of {0,0}. So vector[{2,2}] will return Arrow[{{0,0},{2,2}}], which is a vector starting at the origin and ending at {2,2}. – Lukas Lang Jan 2 at 18:59