Find the volume of a room 15 feet by 16 feet with an 9-foot ceiling.
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Let 
be a rectangular 
matrix with column vectors 
, i.e.
Then we have
and the product of the two is
Because the columns of are orthonormal, we have
which means
reduces to an 
matrix with ones along the diagonal and zero everywhere else, i.e.
where
denotes the identity matrix. This means the solution to 
is given by
Then we have
and the product of the two is
Because the columns of
which means
where