It will be important to compute the set of all vectors that are orthogonal to a given set of vectors. It turns out that a vector is orthogonal to a set of vectors if and only if it is orthogonal to the span of those vectors, which is a subspace, so we restrict ourselves to the case of subspaces.
Taking the orthogonal complement is an operation that is performed on subspaces.
Let be a subspace of Its orthogonal complement is the subspace
The symbol is sometimes read “ perp.”
This is the set of all vectors in that are orthogonal to all of the vectors in We will show below that is indeed a subspace.
We now have two similar-looking pieces of notation:
Try not to confuse the two.
The orthogonal complement of a line through the origin in is the perpendicular line
The orthogonal complement of a line in is the perpendicular plane
The orthogonal complement of a plane in is the perpendicular line
We see in the above pictures that
The orthogonal complement of is since the zero vector is the only vector that is orthogonal to all of the vectors in
For the same reason, we have
From the definition of it would seem that to check whether a vector is in requires us to check infinitely many equations: is for all the different vectors in In practice, we have to check only a finite number of equations.
Let be a subspace of and suppose that A vector in is in if and only if
The next proposition gives us a convenient way of working with and understanding the orthogonal complement of a column space. Later, we will give several shortcuts for computing the orthogonal complements of subspaces presented in other ways–in particular, as null spaces. To compute the orthogonal complement of a general subspace, usually it is best to rewrite the subspace as the column space or null space of a matrix, as in this important note in Section 3.3.
Let be a matrix and let Then
Since column spaces are the same as spans, we can rephrase the proposition as follows. Let be vectors in and let Then
Again, it is important to be able to go easily back and forth between spans and column spaces. If you are handed a span, you can apply the proposition once you have rewritten your span as a column space.
By the proposition, computing the orthogonal complement of a span means solving a system of linear equations. For example, if
then is the solution set of the homogeneous linear system associated to the matrix
This is the solution set of the system of equations
In order to find shortcuts for computing orthogonal complements, we need the following basic facts. Looking back the the above examples, all of these facts should be believable.
Let be a subspace of Then:
See these paragraphs for pictures of the second property. As for the third: for example, if is a (-dimensional) plane in then is another (-dimensional) plane. Explicitly, we have
the orthogonal complement of the -plane is the -plane.
The row space of a matrix is the span of the rows of and is denoted
If is an matrix, then the rows of are vectors with entries, so is a subspace of Equivalently, since the rows of are the columns of the row space of is the column space of
We showed in the above proposition that if has rows then
Taking orthogonal complements of both sides and using the second fact gives
Replacing by and remembering that gives
To summarize:
For any vectors we have
For any matrix we have
As mentioned in the beginning of this subsection, in order to compute the orthogonal complement of a general subspace, usually it is best to rewrite the subspace as the column space or null space of a matrix.
Suppose that is an matrix. Let us refer to the dimensions of and as the row rank and the column rank of (note that the column rank of is the same as the rank of ). The next theorem says that the row and column ranks are the same. This is surprising for a couple of reasons. First, lies in and lies in Also, the theorem implies that and have the same number of pivots, even though the reduced row echelon forms of and have nothing to do with each other otherwise.
Let be a matrix. Then the row rank of is equal to the column rank of
In particular, by this corollary in Section 3.4 both the row rank and the column rank are equal to the number of pivots of
