how to form a differential equation for vector space
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The first vector space you learn about is probably the "arrows" in the xy-plane that go from the origin to any point (x,y). You can add "arrows" together -- e.g. the vector to (a,b) plus the vector to (c,d) is the arrow from the origin to (a+c,b+d). You can scale "arrows" -- e.g. if you want to scale the vector from the origin to (a,b) by the constant c, you get the vector from the origin to the point (ca,cb). These scalable, addable "arrows" satisfy all the requirements for a vector space.
But they aren't the only things that do. Arrows in 3 dimensions from the origin to the points of the form (x,y,z) with the usual ideas of adding and multiplying work too. But so do things that aren't at all like arrows in Euclidean space. As it turns out, the set of all solutions of a linear homogeneous ODE with constant coefficients is one such set that isn't at all like vectors in Euclidean space, but they are a vector space. How does this work?
A solution to the equation is now a vector. Given any two vectors (i.e. solutions) in the space, you can add them together and get a vector in the space. Why? Because the sum of any two solutions must be a solution. (That's a REALLY important property of linear homogeneous odes and isn't too hard to show. You should show it yourself if you haven't seen it already.) Also, given a vector (i.e. a solution) in the space, you can multiply it by a constant and get another vector in the space. Again, that's a really important property of such equations that isn't hard to show.
But they aren't the only things that do. Arrows in 3 dimensions from the origin to the points of the form (x,y,z) with the usual ideas of adding and multiplying work too. But so do things that aren't at all like arrows in Euclidean space. As it turns out, the set of all solutions of a linear homogeneous ODE with constant coefficients is one such set that isn't at all like vectors in Euclidean space, but they are a vector space. How does this work?
A solution to the equation is now a vector. Given any two vectors (i.e. solutions) in the space, you can add them together and get a vector in the space. Why? Because the sum of any two solutions must be a solution. (That's a REALLY important property of linear homogeneous odes and isn't too hard to show. You should show it yourself if you haven't seen it already.) Also, given a vector (i.e. a solution) in the space, you can multiply it by a constant and get another vector in the space. Again, that's a really important property of such equations that isn't hard to show.
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