text: can ever vector in the xy-plane be written as a linear combination of vecto u=91,4) and vector v=(-2,5)?   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: (these are matrices/vectors) |1 -2 | |a|  = |x| |4  5 | |b| ...|y| so a - 2b = x   4a + 5b = y if you reduce this, it becomes a - 2b = x 13b = y - 4x  a is the number multiplying your (1,4) vector b is the number multiplying your (-2,5) vector x is the x coordinate of any vector you want y is the y coordinate  so 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.  what about a? if the same is true for a, then any vector can be written as a linear combination of the two. so again, we had:  a - 2b = x 13b = 2x + y substitute b into the first equation a + 8/13x - 2/13y = x a = 5/13x + 2/13y  so a can also be found given any x,y coordinate. so the answer is yes.  p.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.  p.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.  alternatively, if you don't need to actually find the linear combinations, you can just do it this way: it'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.