Processing, 197 195 bytes

void setup(){size(255,255);int r=255,a=r,g=0,b=0,i,j,k;background(a);for(i=0;i<a;k=i<43?g+=6:i<85?r-=6:i<a/2?b+=6:i<170?g-=6:i<213?r+=6:a>1?b-=6:0,i++)for(j=0;j<a;point(i,j++))stroke(r,g,b,a-j);}

This outputs the image in a 255x255 sized window

Explained

void setup(){            //this is required
  size(255,255);         //size of sketch
  int r=255,a=r,g=0,b=0,i,j,k; //declaring our vars
  background(a);         //set the background colour as white
  //for-loop for the x-coordinates, it also increments/decrements rgb values
  //  based on the x-coordinate
  for(i=0;i<a;k=i<43?g+=6:i<85?r-=6:i<a/2?b+=6:i<170?g-=6:i<213?r+=6:a>1?b-=6:0,i++)
    //for-loop for the y-coordinate (alpha)
    for(j=0;j<a;point(i,j++))   //2) then draw the point at the location
      stroke(r,g,b,a-j);        //1) set the colour of point
}

Edits

• Used int instead of float
• THIS SOLUTION IS INVALID DUE TO NEW RULES BEING ADDED. I WILL UPDATE ANSWER SOON