text: To find out if every vector in the xy-plane can be written as a linear combination of those two, you would do a little linear algebra:\n(these are matrices/vectors)\n|1 -2 | |a|  = |x|\n|4  5 | |b| ...|y|\nso a - 2b = x\n  4a + 5b = y\nif you reduce this, it becomes\na - 2b = x\n13b = y - 4x\n\na is the number multiplying your (1,4) vector\nb is the number multiplying your (-2,5) vector\nx is the x coordinate of any vector you want\ny is the y coordinate\n\nSo you see that b is expressed as b = 1/13y - 4/13x, so no matter what your x and y value is, b will be a number.\n\nWhat about a? If the same is true for a, then any vector can be written as a linear combination of the two.\nSo again, we had:\n\na - 2b = x\n13b = 2x + y\nsubstitute b into the first equation\na + 8/13x - 2/13y = x\na = 5/13x + 2/13y\n\nSo a can also be found given any x,y coordinate. So the answer is Yes.\n\nP.S. I'm a bit tired right now so the math might be a little off. I hope you still get the general idea. My answer of yes, however, doesn't change, but the value of a and b could be off by a little.\n\nP.P.S. You can verify this. Think of any vector you want, and find the value of a and b. Then do a*(1,4) + b*(-2,5). You'll get back your vector.\n\nAlternatively, if you don't need to actually find the linear combinations, you can just do it this way:\nIt's easy to see that (1,4) and (-2,5) are not parallel. Any two non-parallel vectors in the xy plane span the whole plane.