1). The space of solutions to the linear systemAx=0, where

2.) The set of all quadratic polynomials that satisfy

3.) The space of all solutions to the homogeneous ordinary differential equation

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- Jun 23rd 2008, 03:11 PMJCS007Find a basis and the dimension of the following subspaces:
1). The space of solutions to the linear system

*A***x**=**0**, where

2.) The set of all quadratic polynomials that satisfy

3.) The space of all solutions to the homogeneous ordinary differential equation - Jun 23rd 2008, 05:35 PMPaulRS
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.