Find the explicit change of basis matrix from the standard basis of to the orthonormal basis
The change of basis matrix which changes between two orthonormal bases is given by the formula
So,
Find the explicit change of basis matrix from to .
Change of basis matrices between orthonormal bases are orthogonal, so
Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of spanned by the set of vectors,
Applying Gram-Schmidt yields the vectors,
which are all orthogonal. To obtain an orthonormal basis, we divide each vector by its magnitude, and obtain
Let
Find a basis of the subspace of of all vectors perpendicular to
Vectors perpendicular (orthogonal) to are all vectors so that
This equation is the same as,
Which is a matrix equation (there’s only one row, but it’s still a matrix equation):
We’re already in row echelon form, and there are three columns without pivots, with dependent variable,
We play our usual game to find a basis (make all free variables zero except one, repeat, repeat):
If is a subspace of a vector space , prove that is a subspace of .
We follow the same recipe as we always do to prove something is a subspace:
Let , be vectors in . Then we just need to show that is, too. Let be a vector in . Then , meaning that it is indeed in (why?). So is a subspace of .
Find a basis for the kernel of the matrix ,
To find a basis for the kernel, we first row reduce the matrix , to obtain,
We then read off the solutions to the matrix equation (just like finding bases for eigenspaces); there is one pivot and three free variables, so a basis would be,
Find a basis for the column space of the matrix ,
To find a basis for the column space, we column reduce the matrix , to obtain
We can then take the nonzero columns as a basis for the column space: