1). The space of solutions to the linear system Ax=0, where
2.) The set of all quadratic polynomials that satisfy
3.) The space of all solutions to the homogeneous ordinary differential equation
1). The space of solutions to the linear system Ax=0, where
2.) The set of all quadratic polynomials that satisfy
3.) The space of all solutions to the homogeneous ordinary differential equation
1.) Suppose thus:
We must have: note there that we may take and as parameters and write and according to the values of them easily.
Just summing both equations we get; and:
Thus: , are the parameters and note that this can be written as: so any vector in that subspace is a linear combination of and (and they are LI) thus is a base and the dimension of the subspace is 2
2.) Let that polynomial is in our set iff: then consider and as parameters, then where are our parameters
thus any polynomial is in the subspace iff it is a combination of and these 2 are LI, thus is a base of this subspace and its dimension is 2.