Homogeneous Systems The Null Space We have seen that the solution to the homogeneous system of equations Ax = 0 is a subspace of R^{n}. We will now begin a discussion of how to find a basis for this system. The approach we will take is by an illustrative example.
Example Find a basis for the null space of the matrix
Solution We have seen before that the null spaces of row equivalent matrices are the same. Hence this question is equivalent to that of finding the null space of
Now lets rewrite the system in equation form
x_{1} 5x_{3} + 4x_{4}
+ 3x_{5} = 0 We can move the last three variable (the ones that are not corner variables) to the right hand side of the equations and add identity equations to get
x_{1} = 5x_{3}  4x_{4}  3x_{5} It is useful to introduce parameters here s_{1} = x_{3 } s_{2} = x_{4 } s_{3} = x_{5} so that
x_{1} = 5s_{1}  4s_{2}  3s_{3} and we can write this in vector form
We can see that the null space is represented by triplets (s_{1}, s_{2}, s_{3}). This is equivalent (isomorphic) to the space R^{3}. We select the standard basis (1,0,0), (0,1,0), (0,0,1) and come up with the basis for the null space {(5,5,1,0,0), (4,3,0,1,0), (3,2,0,0,1)} Example Let
Find the null space of A.
Solution As before, we find rref the matrix.
The corresponding equations are
x_{1} = 0 and we see that the null space is the subspace containing only 0. Nonhomogeneous Systems Now that we know how to solve the homogeneous equation Ax = 0 we move on to nonhomogeneous systems Ax = b We use the technique of rref as with homogeneous systems. The next example illustrates.
Example Solve
Solution We solve the augmented matrix
and find the rref of the augmented matrix. We get
This gives us the solution
Notice that this is not a vector space (it does not contain the zero vector) so it does not make sense to ask for a basis for a null space. The above answer shows that The solution to Ax = b can be written in the form x = x_{p} + x_{h} Where x_{p} is a particular solution to the nonhomogeneous equation x_{h} represents the null space of A (the solution to the homogeneous equation)
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